chapter two year/mathe… · chapter two(ordinary differential equations) page 1 definition: any...

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Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable) is called a differential equation. : − 1− = + 5 , , !"# $ !". 2− ’’ + 2 ’’ # ( + = cos , ,., !" $ !". 3− ./ . + ./ . = 20. ,. , 0 !", $ !". 1ℎ ,. : − 1 − 3 , 3. ,. #: − ℎℎ !! !" 1# 2#. 2 − 4 , 4. ,. #: − ℎℎ !! 5 ℎ $ !" 4 3#. ,: −1ℎ ,. ℎ ℎ6ℎ !! ℎ ,. . Chapter Two ORDINARY DIFFERENTIAL EQUATION

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Page 1: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 1

Definition: Any equation involving one derivative or more in the unknown

function (the dependent variable) is called a differential equation.

���: −

1 − �� = � + 5 , �� � ������������ ��������, � �� ������� ��������

��� ������ !����"��#��� � �� $��� ������ !����"��. 2 − �'' + 2��''#( + �' = cos � , �� ,. �, � �� �� ������ !����"�� ��� � �� $��� ������ !����"��. 3 − ./

. + ./. = 20. �� ,. �, 0 �� �� ������ !����"��, � ��� � ���

$��� ������ !����"��. 1ℎ��� ��� ��� �� � �� ,. ��: −

1 − 3������� ,����������� �������� �3. ,. �#: −�������� �ℎ��ℎ ��!��!� ���� ��� ���� ������ !����"�� �� ������ �������� ������������ ��������

�� �� �� �1#��� �2#. 2 − 4������ ,����������� �������� �4. ,. �#: −�������� �ℎ��ℎ ��!��!� 5���

�ℎ�� ��� $��� ������ !����"�� ������ 4������ ������������ ��������

�� �� �� �3#. ,���������: −1ℎ� ����� �� � ,. � �� �ℎ� ℎ�6ℎ��� ����� ����!���!� ������� �� �ℎ� ,. �.

Chapter Two

ORDINARY DIFFERENTIAL EQUATION

Page 2: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 2

1ℎ� ��6��� �� � ,. � �� �ℎ� ℎ�6ℎ��� ���� �� �ℎ� ℎ�6ℎ��� ����� ����!���!� ������� �� �ℎ� ,. �. ,��: −7�� �������� ��������6 �ℎ� ,. �, �� ������ � �������� �� ,. �. ��: −$� � = sin 2� , � �������� �� �ℎ� ,. � , �'' + 4� = 0?

=��: −� = sin 2� , �' = 2 cos 2� , �'' = −4 sin 2�, ��"������� 6�!��: − − 4 sin 2� + 4 sin 2� = 0. ∴ � = sin 2� , �� � ��������. ,��: −1ℎ� ��� �� ��� �������� �� � ,. � �� ������ �ℎ� 6������ �������� �� �ℎ� ,. �.

��: −� = �? cos 3� + �( sin 3� , �?, �( ��� ��"����� ���������, �� 6������ �������� �� �ℎ� ,. �, �'' + 9� = 0. ,��: −7�� � ����� �������� �� �ℎ� ,. � 5����� obtained at certain condition,

�� ������ �ℎ� ��������� �������� �� �ℎ� ,. �. ��: −� = sin 3� , � = cos 3� , � = − sin 3� + cos 3� , ��� ���

��������� �������� �� �ℎ� ,. � �'' + 9� = 0. ,��: −1ℎ� ,. � ���ℎ ��′� ��"������ ���������� �� ������ ������� !���� ��"��5

�� ��� �ℎ� ���������� 6�!�� �� ��5� !���� �� �ℎ� ���� ������ !����"��. ��: −1ℎ� ,. � �'' + 3� = � , ���ℎ ��������� ��2# = 3, ��2# = 0 , �� ������� !���� ��"��5 �B. C. D#. ,��: −1ℎ� ,. � ���ℎ ��′� ��"������ ���������� �� ������ "������� !���� ��"��5 �� �ℎ� ���������� ��� 6�!�� �� ��������� !���� �� �ℎ� ���� ������ !����"��. ��: −1ℎ� ,. � �E + 9�' = 0 ���ℎ ��������� ��0# = 3, ��1# = 0 ��

"������� !���� ��"��5 �F. C. D#.

,���������: −

Page 3: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 3

G����� ������������ ��������: −7� ��ℎ ����� ������ ,. � �� �� �������� �� �ℎ�

���5 ∶ − �I��# �J���J + �?��# �JK?�

��JK? ± − − +�J��#� = "��#

�ℎ��� �I, �?, − − − − �J, ��� "��# ��� ��������� �� �. $� 5��� "� ������

�. �. � �ℎ� ������� �������� � ��� ��� ��'�����!���!�. ���5 ��: −

1# �E + 3�M� = sin �, ������ ,. �

2# �E� + �' = 0 , ��� − ������ ,. �

3# �E + 2�� = sin � , ��� − ������ ,. �

N�� �ℎ� ��������6 ,. �� �����5���: −�# �ℎ� �����, "#�ℎ� ��6��� , �#���������

�# ������� �������� ��� ������ !����"��#, �# $��� ������ !����"��, �# 1� �?

1# ��E#( − 3��' + �� = 0. 2# �O�O + ��E' = � . 3# �(�E − �� = 1 − sin �. 4#

�P�P = �( + 1.

5# ��QR�Q#( + �QR

�Q + � �R� = 0.

6# �P�P = �( + 1.

7# ��Q�Q#U

Q + � = �. 8#

.Q/

.Q + ./. ∙ ./

. + .Q/.Q = 0.

H.W 1

Page 4: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 4

7 ������������ �������� �ℎ��ℎ 5�� "� ������� �� �ℎ� ���5

�� = ���, �# − − − − − �1#

�� �ℎ� ���5

X��, �#�� + Y��, �#�� = 0 − − − − − −�2#

Zℎ��� ���, �#, X��, �#, ��� Y��, �#��� ��������� �� ��� !����"�� �, ��� �. 1� ���!� �ℎ� �����[R�\R ������\]R\\ ,. �, �� ℎ�!� �ℎ� ��������6 �����: −

$� �ℎ�� ���� �ℎ� ,. � ��� "� ������� �� �ℎ� ���5 ���#�� + 6��#�� = 0

$���6������ �� "��ℎ ����� 6�!�� �ℎ� 6������ ��������

^ ���#�� + ^ 6��#�� = � , �ℎ��� � �� �� ��"������ ��������. �� 1: −=��!� �ℎ� ,. �, �� + 1# �

� = �?

=��: − �� =

_? ==≫ ^ � = ^ �

_?

ln � = ln|� + 1| + � ==≫ � cd = �cd|_?|_e

� = 7�� + 1# , 7 = �e .

First_ Order First_ Degree Differential

Equation

f − Cghigjklm nlolhgjkl �i. l mloghgpiqr qs tghigjklm#

Page 5: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 5

�� 2: − =��!� �ℎ� ,. � ��2� − 3#�� + ��( + 1#�� = 0?

��( + 1#�� = −��2� − 3#��, ^ �(KM = − ^ �

Q_? ==≫

?( ln |2� − 3| + ?

( ln |�( + 1| = 2� = �?.

�� 3: −=��!� �ℎ� ,. � �� �� + Q_? �� = 0

=��: − �� �� = − (Q_?) �� ==≫ ^ �� �� = − ^ (Q_?)

��

�� − � = − ^ u� + ?v �� ==≫ �� − � = −(Q

( + ln � ) + � y �

1 �

0 �

N��� �ℎ� 6������ �������� ��� ���ℎ �� �ℎ� ��������6 ,. ��: −

1) �� = O

(KM) , 2) �((� + 1) �� + �((� − 1)�� = 0

3) sin � �� + cosh � = 0, 4) ln � �

� = .

5) �((�( + 1)�� + � √�( + 1 ��, 6) �(� �� = (1 + �) csc �.

7 �������� �� �(�, �)�� ���� �� "� ℎ�5�6����� �� ��6��� � �� �(y�, y�) =yJ�(�, �)

Y�� , �(�, �) = �( + 3�� + �(, �(y�, y�) = y(�( + 3y�. y� + y(�(

= y((�( + 3�� + �() = y( �(�, �). ∴ �(�, �)�� � ℎ�5�6����� �������� �� ��6��� ���.

H.W 2

z − {q|q}lrq~m �isslhlrpigk ��~gpiqrm

Definition:

Page 6: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 6

1ℎ� ���� − ����� − ����� − ��6��� ,. � ��� "� �� �� ∶ �� = ���,#

�Q�,# − − − − − −�1#

$� "��ℎ �?��, �#, �(��, �#��� ℎ�5�6����� ��������� �� �ℎ� ��5� ��6��� �ℎ�� �ℎ� ,. � �� ���� �� "� ℎ�5�6�����. $� ��� "� ������� �� �ℎ� ���5: −

�� = N u

v − − − − − �2#

1� ���!� �� �� ! = ==≫ � = ! � ==≫ �

� = ! + � ���,

��"������� �� �� �2# 6�!��: −

! + � ��� = N�!# ==≫ N�!# − ! = � ��

�����#K� = �

��� ���"�� ,. �#

$���6������ �� "��ℎ ����� 6�!�� �ℎ� ����� �������� ^ �����#K� = ln |�| + �.

���: −N��� �ℎ� 6������ �������� ��� ���ℎ �� �ℎ� ��������6 ,. ��: −

1) (�M + �M)�� − 3��( �� = 0?

=��: �� = U_U

MQ ==≫ �� = �U (U_U)

�U(MQ) , ∴ ,. � �� ℎ�5�6.

,�!����� "� �M 6�!�� ==≫ �� = ?_u�

�vU

Mu��vQ , ==≫ ! + � ��

� = ?_�UM�Q

� ��� = ?_�U

M�Q − ! ==≫ � ��� = ?_�UKM�U

M�Q , ==≫ � ��� = ?K(�U

M�Q (�� ,. �)

^ M�Q?K(�U �! = ^ �

==≫ − ?( ln |1 − 2!M| = ln |�| + �

ln �1 − 2 uvM� + 2 ln|�| = �? , �ℎ��� �? = −2�

Page 7: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 7

2# u���� + �v �� − � �� = 0?

=��: − �� = \�

�_ ==≫ �

� = �\����_�� �ℎ�5�6. ,. �#

,�!����� "� � 6�!��:- �� = �/ +

, ==≫ ��� ! = , ==≫ � = !. �, ==≫ �

� = ! + ��!/��

! + � ��� = �� + ! ==≫ � ��

� = �� + ! − ! ==≫≫ � ��� = ��

^ �K� �! = ^ � ==≫ −�K� = ln |�| + � ==≫ ln |�| + �K� = �? ,

ln |�| + �K�� = �? (�ℎ� ��������) .

3) u2� sinh + 3� cosh

v �� − 3� cosh �� = 0?

=��: − �� = ( ��d��

�_M ������

M ������

, ℎ�5�6. ,. � ,�!���� "� � ��� ��"���. ��� ! =

, ��� ! + � ��� = �

� ==≫

! + � ��� = ( ��d� �_M� ���� �

M ���� � , ==≫ � ��� = ( ��d� �

M ���� �

^ ���� ���d� � �! = (

M ^ � ==≫ ln �sinh

� − (M ln � = �.

�?� + "?� + �?)�� + (�(� + "(� + �()�� = 0, "� ��������5�����

� = � + ℎ, � = � + �, �� �ℎ��ℎ � = ℎ, � = �, ��� �ℎ� ��������

�� �ℎ� �������� (�?� + "?� + �?) = 0 , ��� (�(� + "(� + �() = 0

���: −=��!� �ℎ� ,. ��: −

1) (� − � − 1)�� + (4� + � − 1)�� = 0

=��: − � − � − 1 = 04� + � − 1 = 0} ��5����������� ���!�� ==≫ −5� = 0,

==≫ � = 0, ��� � = 1, �� � = ℎ = 1, ��� � = � = 0.

� − ��l l�~gpiqr (gf� + jf� + �f)�� + (gz� + jz� + �z)�� = �

im hlkgpl� pq p�l �q|q}lrq~m sqh|

Page 8: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 8

1ℎ� ��������5����� ∶ � = � + ℎ ==≫ � = � + 1 , �� = ��

� = � + � ==≫ � = � , �� = ��

Y�� ,�� = − KK?

O_K? , �� 6�� ���� = − �_?K�K?

O�_�_?K? =− �_�

O�_� , ℎ�5�6. ��. ��� ! = �� , � = !. �, ��

�� = ! + � ���� , ��"��.

! + � ���� = − ?K�

O�_? ==≫ � ���� = − ?K�

O�_? − ! = − ?_O�QO�_?

^ O�_??_O�Q �! = − ^ ��

� ==≫ ?( ln |1 + 4!(| + ?

( tanK?(2!) = − ln |�| + �

?( ln |1 + 4 u�

�v( | + ?( tanK? u2 �

�v + ln |�| = �. 2) (2� + 2� + 1)�� + (2� + � − 2)�� = 0?

=��: − 2� + 2� + 1 = 02� + � − 2 = 0 } ���!� ��5���. ==≫ � + 3 = 0 ==≫ � = −3

� = ¡M , �ℎ�� � = ℎ = ¡

M , � = � = −3

1ℎ� 1�������5����� ∶ � = � + ℎ = � + ¡M , �� = ��,

� = � + � = � − 3, �� = ��, ��� ���� = − ((�_(�)

(�_� = − (_Q¢£

(_¢£

1ℎ� ��5� ��� � �� "����� � ���� = −(2 + !), ^ ��

(_� = − ^ ��/�

ln |2 + !| = − ln |�| + �, ln |2 + �� | + ln |�| = �.

N��� �ℎ� 6������ �������� ��� ���ℎ �� �ℎ� ��������6 ,. ��: −

1) �( �� + (�( − ��)�� = 0. 2) (� + �)�� + (� − �)�� = 0. 3) �' =

+ cos(K ) .

4) � �� − � = ¤�( − �(.

5) (�� + �()�� = (�( + � � + �()��. 6)

�� = K_¡

_K? . 7) (2� − 5� + 3) �� − ( 2� + 4� − 6)�� = 0. 8) (�( + �()�� + �� �� = 0.

H.W 3

Page 9: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 9

1ℎ� �������� �� + 4��#� = ¥��# − − − −�1#

�� �� + 4��#� = ¥��# − − − −�2#

�ℎ��� ���� 5��"�� �� ������ �� "��ℎ �ℎ� �� ������ !����"��� ��� ��� ����!���!�

�� ������ � ������ �������� �� �ℎ� ����� �����. Zℎ��� 4 ��� ¥ ��� ���� ������� ���������.

X���� �� "��ℎ ����� �� �1#"� � �������� $��#�� "� ����� ����� ���ℎ �ℎ��

�� �$. �# = $. �

� + 4��#. $. �

$. �� + �. �¦

� = $. �� + 4. $. � ==≫

�¦� = 4. $ ==≫ ^ �¦

¦ = ^ 4. �� ==≫

ln $ = § 4 �� ==≫ $ = �^ ¨ � , $��#�� ������ $���6�����6 ������ �� ,. �

Y�� , �� �$. �# = $. ¥��#, ����6������ 6�!��

$. � = ^ $� �# ¥��#��

�ℎ��ℎ �� �ℎ� 6������ �������� �� ������ ,. �. ���: −=��!� ���ℎ �� �ℎ� ��������6 ,. ��: −

1# �� + 2�� = 4�, ������ ,. � , 4��# = 2�, ¥��# = 4�

$��# = �^ ¨�#� ==≫ $��# = �^ ( � = �( �� �� ����6�����6 ������

Y�� , $. � = ^ $. ¥��# �� ==≫ �(. � = ^ �( . 4� ��

1ℎ�� , �(. � = 2 �( + � ==≫ � = 2 + � �K( ��ℎ� ��������#.

© − ªirlgh �isslhlrpigk ��~gpiqr

Page 10: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 10

2# � ���� = � + �M + 3�( − 2�, � ��

�� − � = �M + 3 �( − 2� ]��!���� "� �

���� − �

� = �( + 3� − 2 , ������ ,. �, ��� 4��# = 1� , ��� ¥��# = �( + 3� − 2

$ = �^ ¨�#� = �K ^��� = �K cd = ?

, �� �� $���6�����6 ������. 1ℎ�� , $. � = ^ $. ¥��#�� ==≫ ?

. � = ^ ? . ��( + 3� − 2#��

= ^ u� + 3 − (

v �� = Q( + 3� − 2 ln � + �

� = U( + 3�( − 2� ln � + � �.

3# �� + � cot � = 5 ���� , ���� �ℎ� ��������� �������� 6�!�� � u¬

(v = −4. =��: − 4��# = cot � , ¥��# = 5 ����

$ = �^ ¨�#� = �^ ��­ � = �cd ��d = sin � , ��� $. � = ^ $ . ¥��#��

sin � . � = 5 ^ sin � . ���� �� = −5 ���� +c

����� � u¬(v = −4 , �ℎ�� − 4 sin ¬

( = −5 ����®Q + �

−4 = −5 + �, ==≫ � = 1, �ℎ� ��������� �������� ��

� ∙ sin � = −5 ���� + 1 , �� � sin � + 5 ���� = 1

$� ℎ�� �ℎ� ���5 , �� + 4��# � = ¥��# �J − − − − − �1# � ≠ 1

�� ���� + 4��# � = ¥��# �J − − − −�2# � ≠ 1

Y���: − ��� � = 1 , �ℎ� ,. �. "���5�� �� ���"�� ,�!����� �� "��ℎ ����� �� �� �1#"� �J 6�!��: �KJ �

� + 4��# �?KJ = ¥��#, ��� 0 = �?KJ ==≫ �/� = �1 − �# �KJ �

��� "� ��"��������� , ??KJ

�/� + 4��# 0 = ¥��#

° − Flhrqkki'm �isslhlrpigk ��~gpiqrm

Page 11: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 11

�/� + �1 − �#4��# . 0 = �1 − �# ¥��# , �ℎ��ℎ �� ������ �. �. � 0 ��� 0′ ���: −=��!� 1# �

� − � = � �¡ , �±�������'� ,. �. #

,�!����� "� �¡ ==≫ �K¡ ���� − �KO = � , ��� 0 = �KO ==≫ �0

�� = −4 �K¡ ����

������� �ℎ� �������� �� − ?O

�/� − 0 = � ==≫ �/

� + 40 = −4�

������ �� 0 ��� 0'

4��# = 4 , ¥��# = −4� , $ = �^ ¨�#� = �^ O � = �O, $. 0 = ^ $ . ¥��# �� , �O. 0 = − ^ �O . 4� �� ==≫

�O . 0 = − ²� �O − ?O �O³ + � ==≫ ?

´ = −� + ?O + � �KO.

2# ��6�( − � − 1#�� + 2� �� = 0

2� �� − ��� + 1# = −6 �M �±�������'� ,. �#

�KM �� − �_?#

( �K( = − M , ��� 0 = �K( =≫ �/

� = −2�KM. ��

��"������ ��� ���!� �� 6�� � � �K( = 6 � + �.

=��!� �ℎ� ��������6 ,. ��: −

1# � �� + 2� �� = �� − 2#��� ?

2# �( �� + 2�� �( − �#�� = 0?

3# � �� + ��� + � − 3�#�� = 0?

4# � �� + 3� = ��d

Q ?

5# cosh � �� + �� sinh � + � # �� = 0?

6# �� + �tan � # � = sin 2� ? , ��0# = −2

7# � �� + � = �M �µ? 8# � �

� − � �( + � = 0? 9# 2 �

� − + �M cos � = 0 ? , 10# � �

� = 2� + �M �? , ��1# = 0

H.W 4

Page 12: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 12

1ℎ� ,. � X �� + Y �� = 0 �� ���� �� "� ����� �� �ℎ��� �� � ��������

N��, �#������ ���ℎ �ℎ�� , �N = X�� + Y �� − − − −�1#

¶���� �N = 0 ==≫ N��, �# = �������� �ℎ��ℎ �� �ℎ� 6������ ��������

Y�� , �N = .�. �� + .�

. �� − − − −�2#

·�5 ������ "������ �� �1#& �� �2# 6�!���

.�

. = X ==≫ .Q�. . = .¹

. .�. = Y ==≫ .Q�

. . = .º. , so

. = .º. − − − −�3#

�� �3#�� � ��������� ��� �ℎ� ,. �. X �� + Y �� = 0 �� "� ����� Y�� .�. = X ==≫ N = ^ X �� + � − − − − − �4#

��"��������� �� �� �4#�� .�. = Y ��� �ℎ�� ����6���� 6�!�� �

ℎ���� N��, �# = · �� �ℎ� 6������ �������� 7� ���5 �� ��� �ℎ�� �� �����6 ∶ − �� �� �������� �ℎ� �������� ���, �# = 3�( �( + 2�M� + �( + · = 0, �ℎ�� �� �� ������������� �ℎ�� ��, 6��

�N = 6�(� �� + 6�(� �� + 2 �M �� + 6�(� �� + 2� �� = 0

"� �� ������6 �ℎ� ������������ �� �� ��� �� , �ℎ��

�6�(� + 2�M#�� + �6�(� + 6�(� + 2�#�� = 0

X = .�. Y = .�

.

. = 12 � � + 6 �( , .º. = 12 � � + 6 �(

» − ��g�p �isslhlrpigk ��~gpiqr

Page 13: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 13

Y�� �� ���!� �ℎ�� ����� ,. �. X = .�

. ==≫ ¼N = X¼� , ��� "� ����6�����6 N = ^ X �� + �

�� ���� � ������������� �ℎ�� N �. �. � �

.�

. = .. �^ X ��# + .e�

. , ����� .�. = Y

.. �^ X ��# + .e�

. = 6�(� + 6�(� + 2�

.. �^�6 �(� + 2 �M# �� + .e�

. = 6�(� + 6�(� + 2�

.. �3�(�( + 2 �M�# + .e�

. = 6�(� + 6�(� + 2�

6�(� + 6�(� + .e�. = 6�(� + 6�(� + 2�, ∴ .e�

. = 2�

^ ¼ � = ^ 2 � ��, ==≫ � = �( + ·

N��, �# = 3�(�( + 2 �M� + �( + ·,�ℎ��ℎ �� �ℎ� ��5� �������� �� ����� ���ℎ ��

��� ��� ��������� ��� ����� ,. � ��

�� .¹. = .º

. , �ℎ�� ��7·1 ,. �., �ℎ�� �ℎ���� ��� ��� �� ���!� 7# ���, �# = ^ X��, �# �� + �

.��,#. = Y , �� 6�� .e�

. , �ℎ�� ���� ����6���� �� 6�� �

±# ���, �# = ^ Y��, �# �� + � , .��,#. = X , �� 6�� .e�

. ,�ℎ�� ���� ����6���� �� 6�� � ���: −=��!� ���ℎ �� �ℎ� ��������6 ,. ��

1# �� + �#�� + �� + �(#�� = 0?

���: − X��, �# = � + � =≫ .¹. = 1, Y��, �# = � + �( =≫ .º

. = 1

Page 14: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 14

. = .º. , ∴ ����� ,. �

1ℎ� �������� �� N��, �# = � �ℎ��� .�. = X , .�

. = Y

Y�� ^ ¼N = ^�� + �# �� =≫ N = Q( + �� + �

.�

. = 0 + � + .e�. = Y = � + �(, �� .e�

. = �(, "� ����6���

^ ¼� = ^ �( ¼� =≫ � = UM + �

�ℎ� �������� �� N��, �# = Q( + �� + U

M + �

2# �3 �M . � − 2�#�� + ��M + 1#�� = 0

=��: − X = 3 �M� − 2� =≫ .¹. = 3 �M

Y = �M + 1 =≫ .º. = 3 �M , .¹

. = .º. , ∴ ����� ,. �

1ℎ� �������� N��, �# = � �ℎ��� .�. = X , .�

. = Y Y�� ^ ¼N = ^�3 �M . � − 2�# �� =≫ N = �M . � − �( + �

.�

. = Y = �M + .e�. = �M + 1, �� .e�

. = 1, ^ ¼� = ^ 1 ¼�

∴ � = � + � , �ℎ� �������� �� N��, �# = �M . � − �( + � + � 3# �cos � + � cos �#�� + �sin � − � sin �#�� = 0?

4# ue¾ + ln y + À¾v dx + u¾

À + ln x + sin yv dy = 0?

$� �� �1#�� ��� ����� , �� ����6�����6 ������ �� ����

7# $� ¼X ¼�⁄ − ¼Y ¼�⁄Y = ���#, � �������� �� � �����, �ℎ��

$ = �^ ��#� , �� �� ����6�����6 ������ �� �1#

Brpl}hgpir} Äg�pqhm

Page 15: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 15

��: −=��!� 3. ,. � ��( + �( + �#�� + � � �� = 0? X = �( + �( + �, ==≫≫ .¹

. = 2� , Y = �� , ==≫ .º. = �

�ℎ�� .¹. ≠ .º

. , �� �� �� ��� �����

1ℎ�� .¹ .⁄ K.º .⁄º = (K

= ? = ���#, $ = �^�

� � = �

$���������6 �ℎ� ����6�����6 ������

��M + � �( + �(#�� + �( � �� = 0 , .¦¹. = 2�� , .¦º

. = 2�� , ∴ .¦¹. = .¦º

.

∴ ����� , �ℎ�� �������� �ℎ� ��5� �������� �� ���J

±# $� .¹ .⁄ K.º .⁄¹ = −6��#, � �������� �� � �����,

1ℎ�� = �^ ]�#� , �� �� ����6�����6 ������ �� �1# ��: − =��!� �2� �O � + 2��M + �#�� + ��(�O� − �(�( − 3�#�� = 0

. = 8��M� + 2��O� + 6��( + 1, .º. = 2��O� − 2��( − 3

�� .¹. ≠ .º

. , �ℎ��ℎ �� ��� ����� , .¹ .⁄ K.º .⁄¹ = ÅU\�_ÅQ_O

( ´ \�_(U_

= O�(U\�_(Q_?#�(U\�_(Q_?# = O

= −6��#

∴ $��# = �^ ]�#� = �K ^´�� = �KO cd = �cd Æ´ = ?

´

X���� �� �ℎ� ,. � "� $��#, �� 6��

u2��O� ?´ + 2� U

´ + ´v �� + uQ´\�

´ − QQ´ − 3

´v �� = 0

u2�� + 2 + ?

Uv �� + u�(� − QQ − 3

´v �� = 0

.¦¹. = 2�� − (

Q − M´ , .¦º

. = 2�� − (Q − M

´ , ∴ .¦¹. = .¦¹

. =

Page 16: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 16

1ℎ�� ���!�� ���ℎ �ℎ� ��5� �������� �� ����� ,. �. ��������

N��� �ℎ� 6������ �������� ��� ���ℎ �� �ℎ� ��������6 3. ,. �: −

1# �2�� + �(#�� + ��( + 2��#�� = 0?

2# Ç� + ¤�( + 1È�� − É� − ¤Q_?Ê �� = 0?

3# �� + � + 1#�� − �� − � + 3#�� = 0?

1# = ����� 1� �� �� 2�� ����� ,. �

2# ¶�5�6����� 2�� ����� ,. �

3# Y�� ¶�5�6����� 2�� ����� ,. �

Ë# ��l l�~gpiqr �gm p�l sqh| qs: Ä u�, ���� , �z�

��zv = �

$� ��� "� ������� �� 1ÌÍ ����� ,. � "� ��"������ , �� "� 4,

��� = ���� , �ℎ�� �

�� = �(���( , �ℎ�� ��"�. ��� ��

�� , �(���( �� �ℎ� 5���� ��.

"� & � �� �� 6�� 1�� ����� ,. � ��� ���!�� "� �ℎ� 5��ℎ��� ���� "�����

�� 6�� ��� �ℎ�� ������ �� ���� �� ���!� "� �� ������� �� !����"�� .

H.W 5

Second Order Differential

Equations

1- Special types of 2nd

Order

Page 17: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 17

��1: −=��!� �ℎ� ��������6 ,. � �'' + ��' = 0

�Q�Q + � �

� = 0, →→ N u�, �� , �Q

�Qv

��� = �� , �ℎ�� �Ï

� = �Q�Q , →→ �Ï

� + � = 0 ] ∗ �Ï

�ÏÏ + � �� = 0, →→ ^ �Ï

Ï = ^ � �� = ^ 0 , →→ ln + Q( = �

�� = �? . �K�Q

Q , →→ ^ �� = ^ �? . �K�QQ �� , →→ � = �? ^ �K�Q

Q �� + �(

F# ��l l�~gpiqr �gm p�l sqh| qs ∶ Ä u�, ���� , �z�

��zv = �

1ℎ�� ��� = �� , �Ï

� = �Q�Q = �Ï

��� = �Ï

� , �ℎ�� ��"�. �� 5��� ��. ��� �ℎ�� � = ^ �� + �

�� 2: − �Q�Q + �

� = 0, ���!�� "� �7#&�±#

���J: − 1# ����� "� ±

��� = �� , �Ï

� = �Q�Q = �Ï

��� = �Ï

� , �ℎ�� ��"��. �� 5��� ��.

u �Ï� + = 0v ∗ �

Ï , →→ � + �� = 0, →→ ^ � + ^ �� = ^ 0 , →→ + � = �

∴ = � − �, →→ �� = � − �, →→ ^ �

eK = ^ �� , →→ − ln�� − �# = � + �?

ln�� − �# = −� − �? , ���� � , � − � = �K . �Ke� , ��� �( = �Ke�

∴ � = � − �( �K

2J� "� 7# ��� = �� , �Q

�Q = �Ï� , �� �ℎ� �� . "� �Ï

� + = 0 �ÏÏ = −�� , →→ ^ �Ï

Ï = − ^ �� , →→ ln = −� − � , = �KKe

= �? �K , →→ �� = �? �K , ^ �� = ^ �? �K �� ,

Page 18: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 18

� = −�?�K + � , � = � − �? �K

Ñ# ��l l�~gpiqr �gm p�l sqh| qs: Ä u�� . ��

��v = �

1ℎ� ����� ����� ������������ �������� ��� �� ������ �� �ℎ� ���5

�� = � u

v , 1ℎ� ������ �����6 ��. ��� 2J� ����� �� � �Q�Q = �? u

, ��v

1ℎ� �������� �� "� ����5��6 = ! →→ � = !� , �

� = � + ! ���

�Q�Q = ��

� + � , �Q��Q + ��

� = � �Q��Q + 2 ��

� , ∴ � �Q�Q = �( �Q�

�Q + 2� ���

∴ �( �Q��Q + 2� ��

� = �( u!, ! + � ���v 3Ò � �Q

�Q = �M u!, � ���v

1ℎ�� ��� ������ ��"��. � = �Í , � = ln � , �Í� = ?

, �QÍ�Q = − ?

Q

��� = ��

�Í . �Í� = ?

. ���Í , ∴ � ��

� = ���Í , �Q�

�Q = − ?Q

���Í + ?

� u���Ív

�Q��Q = − ?

Q���Í + ?

��Í u��

�Ív �Í� = ?

Q�Q��ÍQ − ?

Q���Í

�( �Q��Q = �Q�

�ÍQ − ���Í , �� � �Q

�Q = � u���Í , !v 3Ò �Q�

�ÍQ = � u!. ���Ív.

Zℎ��ℎ ����� "� ���!� ����6 ���� ± ��������

�� 3: −=��!� 2J� 3. ,. �. 2 �(� �Q�Q + �( = �( u�

�v(

=��J: − ��!��� �� "� 2��, �� 6�� � �Q�Q +

( = ( u�

�v( − − − 1

∴ � �Q�Q = � u

, ��v , �� ��� ! =

, ��� � = ! . � − − − − − 2

�� = ! + � ��

� − − − − − 3

�Q�Q = ��

� + � , �Q��Q + ��

� = � �Q��Q + 2 ��

� , ∴ � �Q�Q = �( �Q�

�Q + 2� ��� − − − −4

Y�� ��"��. ��� �2#, �3# , & �4# �� �� �1#

Page 19: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 19

�( �Q��Q + 2� ��

� + �( = ?

(� u! + � ���v(

�( �Q��Q + 2� ��

� + �( = ?

(� Ó!( + 2!� ��� + u� ��

�v(Ô

�( �Q��Q + � ��

� = ?(� u� ��

�v( − − − − − 5

Y�� ��� 2J� ��"������� "� ���� � = �Í , ��� � = ln � , �Í� = ?

, �QÍ�Q − ?

Q

��� ��� = ��

�Í . �Í� = ?

���Í , �ℎ�� � ��

� = ���Í − − − − − 6

�Q��Q = − ?

Q���Í + ?

� u���Ív = − ?

Q���Í + ?

�Í u���Ív . �Í

�Q��Q = ?

Q�Q��ÍQ − ?

Q���Í , ∴ Q�Q�

�Q = �Q��ÍQ − ��

�Í − − − − − 7

Y�� ��"������� �� �6#, �7# �� �� �5#, �� 6��, �Q��ÍQ − ��

�Í + ���Í = ?

(� u���Ív( , ∴ �Q�

�ÍQ = ?(� u��

�Ív(

Y�� ��� = ���Í , �Q�

�ÍQ = �Ï�Í = �Ï

�����Í , ∴ �Q�

�ÍQ = �Ï��

��� = ?

(� (, ��� �� ≠ 0 → �Ï�� = ?

(� → ^ �ÏÏ = ^ ?

(� �!

ln = ?( ln ! + ln �? , → = �? !?/( , ��

�Í = �? !?/( , ^ !K�Q �! = ^ �? ��

!?/( = e�Í( + eQ

( , ∴ ! = ue�Í( + eQ

( v( , = ue�Í

( + eQ( v(

∴ � = �� ·? ln � + ·(#(, �ℎ��� ·? = e�( , ��� ·( = eQ

(

Y�� �� = 0 , �ℎ�� �!�� = 0, § �! = § 0 , ! = 7 , ��� �

� = 7 � = 7� .

Page 20: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 20

�� 4: −���!� �(�'' + 3��' − 3� = 0

���J: − �(�'' + 3��' − 3� = 0] ? , �ℎ�� ��'' + 3�' − 3

= 0

�ℎ��ℎ �� � u , �

� , �Q�Qv , �� � = !� , ��� �

� = ! + � �!/��

��� �Q�Q = ��

� + � , �Q��Q + ��

� = � �Q��Q + 2 ��

� , ∴ � �Q�Q = �( �Q�

�Q + 2� ���

Y�� ��"��. �� 5��� ��. �( �Q��Q + 2� ��

� + 3! + 3� ��� − 3! = 0

�( �Q��Q + 5 � ��

� = 0

Y�� ��� � = ln � , �� = ?

, �QÍ�Q − ?

Q ��� ��� = ��

�Í . �Í� = ?

���Í , �ℎ�� � ��

� = ���Í

��� �Q��Q = − ?

Q���Í + ?

� u���Ív = − ?

Q���Í + ?

u ��Í u��

�Ív . �Í�v

�Q��Q = ?

Q �Q��ÍQ − ?

Q���Í , ∴ Q�Q�

�Q = �Q��ÍQ − ��

�Í

�Q��ÍQ − ��

�Í + 5 ���Í = 0 , �� �Q�

�ÍQ + 4 ���Í = 0

��� = ���Í , & �Q�

�ÍQ = �Ï�Í , �� �Ï

�Í + 4 = 0, ��� ^ �ÏÏ = ^ −4 ��

∴ ln = −4� + ln � , → ln − ln � = −4� , → ln Ïe = −4�

∴ = � �KOÍ , ���Í = � �KOÍ , → ^ �! = � ^ �KOÍ �� , ! = − ?

O � �KOÍ + �?

= − e

O �KO cd + �? , ∴ = 7 �KO + ± , ��� � = 7 �KM + ±

=0�!� �ℎ� ��������6 2J� 3. ,. �. 1# �'' = 1 + �'(, 2# ��'' + �' = 0 , 3# �(�'' + ��' = 1

4# �(�'' + � �' − � = 0 , 5# �(�'' − 2��' + 2� = �M cos �

H.W 6

Page 21: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 21

1ℎ� 6������ �������� ��� "� �� ������ �� �ℎ� ���5: 4 �Q

�Q + ¥ �� + Ò � = Õ��# − − − − − �1#

Zℎ��� 4, ¥, Ò, ��� �������� ������������ �� ��������� �� �

G����� �������� ���ℎ �������� ������������. 1ℎ� ��5� �������� �1#"�� 4, ¥, Ò, ��� ���������, ��� �ℎ� �������� ��: G�� ���# = ���# + !��# − − − − − �2#

�� = �Ö

� + ��� − − − − − −�3#

�Q�Q = �QÖ

�Q + �Q��Q − − − − − −�4#

4 �Q�Q + ¥ �

� + Ò � = u4 �QÖ�Q + ¥ �Ö

� + Ò �v + u4 �Q��Q + ¥ ��

� + Ò !v = Õ��#

4 �QÖ�Q + ¥ �Ö

� + Ò � = 0 = �e �ℎ��ℎ �� ��5 ��5����� �������� − − − �5#

7�� 4 �Q��Q + ¥ ��

� + Ò ! = Õ��# = �Ï �ℎ��ℎ �� ������� �������� − − − �6#

1ℎ�� �ℎ� 6������ �������� �� ∶ � = �e + �Ï − − − − − −�7#

1ℎ� ��5 ��5������ �������� �� �ℎ� ��5 ��5������ ��������

4 �Q×�Q + ¥ �×

� + Ò �e = 0 − − − − − �5Ø#

1��� �� = , �ℎ��ℎ �� ������ �ℎ� , − � ������

∴ ,� = �� , ,,� = ,(� = �Q

�Q , ,M� = �U�U , … ….

Ú�4 ,( + ¥ , + Ò#�e = 0] ?Ï , →→ u,( + Û

¨ , + ܨv �e = 0

2- 2nd

O.H.D.E with Constant

Coefficients

Page 22: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 22

�,( + �?, + �(#�e = 0 − − − −�8#

�� �8#����� "� ������� �� �ℎ� ���5 �� !���"�� 5 ∶ 5( + �?5 + �( = 0 3� �5 − �?#�5 − �(# = 0

1ℎ�� �, − �?#�, − �(#�e = 0 − − − − − − − �9#

,� − �?� = 0 , �Ö� − �?� = 0 − − − − − −�10Ý#

Y�� �ℎ� �������� ��!���� ���� �ℎ��� �����: �# ·��� 1; $� �? ��� �( ��� ��� ����� ��� ���� Y[ , �? ≠ �(

�� ℎ�!� �Ö� − �?� = 0 , �Ö

� = �?� →→ �ÖÖ = �? �� →→ ^ �Ö

Ö = ^ �? ��

ln � = �?� + ln 7 , →→ � = 7 �R� − − − −�11#

�, − �(#�e = � , →→ ��e�� − �(�e = 7 �R�, �ℎ��ℎ �� ������ ���!�� "� $ ������

$� = �^ ¨� = �^ KRQ� = �KRQ

�e = ?¦� ^ ¥��# $� �� = ?

\ÆßQ� ^ 7 �R� . �KRQ �� = ?\ÆßQ� ^ 7 ��R�KRQ# ��

�e = ?\ÆßQ� ² ?

R�KRQ 7 ��R�KRQ# + ±³ , ��� ?

R�KRQ7 = 7′

�e = ?\ÆßQ� Ú7'�R� . �KRQ + ±]

∴ �e = 7'�R� + ± �RQ − − − − − �12#

∴ �e = 7 �R� + ± �RQ , �� �ℎ� ��5 ��5������ �������� ��� �ℎ� ���� 1

"# $� �? = �( ��� ���� ��5� �� ���J �11# , � = 7 �R� − − − − − �11#

�, − �(#�e = 7 �R� , �×� − �( �e = 7 �R� , �ℎ� ��5� �������� "� $������

�e �KRQ = ^ 7 ��R�KRQ# �� & �? = �(, ∴ �? − �( = 0

Page 23: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 23

�e �KRQ = ^ 7 �I �� = 7 � + ± , →→ �e = �7 � + ±#�RQ − − − −�13#

�e = �7 � + ±#�RQ , �� �ℎ� ��5 ��5������ �������� ��� ���� 2. �? = �(. �# $� �ℎ� ����� �ℎ��� $5�6�����: − �? = à + � á , �( = à − � á , �ℎ� �������� �� �ℎ� ��5� �� ���J�12#

�e = 7 �R� + ± �RQ = 7 ��â_ãä# + ± ��âKãä# = �â�7 �ãä + ± �Kãä#

7�� "� �ãå = cos æ + � sin æ , �� �ãä = cos á� + � sin á� , ��� �Kãä = cos á� − � sin á�

�� �e = �âÚ7�cos á� + � sin á�# + ±�cos á� − � sin á�#] �e = �âÚ�7 + ±# cos á� + �7 − ±# sin á� ]

�e = �â�7' cos á� + ±' sin á�# − − − − − �14#

Zℎ��ℎ �� �ℎ� ��5 ��5������ �������� ��� 3R� ���� �ℎ��� �ℎ� ����� �5�6�����

Y���: �e �� �ℎ� 6������ �������� �� Õ��# = 0, �ℎ��ℎ �� ������ ������ ¶. ,. �. �� 1: −=��!� �ℎ� ,. � �Q

�Q − 6 �� + 9 � = 0

=0�J: − �Q�Q − 6 �

� + 9 � = 0, �ℎ��ℎ �� 2J� 3���� ������ ℎ�5�6. ,. �

�,( − 6 , + 9#� = 0 , �5( − 6 5 + 9# = 0, �5 − 3#�5 − 3# = 0

∴ �? = �( = 3 , �ℎ��ℎ �� ���� 2, �� � = �7� + ±#�M

�� 2: −���!� �Q�Q + 5 �

� + 4� = 0

�,( + 5, + 4#� = 0 , �5( + 55 + 4# = 0, �5 + 4#�5 + 1# = 0

∴ �? = −4 & �( = −1 , �� �? ≠ �( �ℎ��ℎ �� ���� 1.

Page 24: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 24

�� 3: −���!� �ℎ� ,. � �Q�Q + 2 �

� + 2 � = 0

���J: − �,( + 2, + 2#� = 0 , �5( + 25 + 2# = 0 , 5 = K(± √OKÅ( = −1 ± �

�? = à + �á & �( = à − �á , ��� ����� �5�6����� ∴ ���� 3

à = −1 , á = 1 , � = �K�7 cos � + ± sin �#

1ℎ� ��������� �������� Õ��# ≠ 0��� ��Ï#

4 �Q�Q + Õ �

� + Ò� = Õ��# − − − −�1#

u4 �QÖ�Q + Õ �Ö

� + Ò �v + u4 �Q��Q + Õ ��

� + Ò!v = Õ��# − − − −�1Ø#

4 �QÖ�Q + ¥ �Ö

� + Ò� = 0 = �e , �ℎ��ℎ �� ��5 ��5������ �������� − −�5#

��� �ℎ� 6������ �������� �� � = �e + �Ï − − − − − �6#

�ℎ��� �e �� ����� �� "����� �ℎ�� Õ��# = 0. 1ℎ� ��������� �������� Ç�ÏÈ �� �"������ "� ����6 5��ℎ�� �� �������5���� ����������� �ℎ��ℎ �� � ����� ��� ����� �������� ��� �� �� ���� �� � ��. Case (a) : If ç��# = �qrmp. = �f

4 �Qè�Q + ¥ �è

� + Ò �Ï = �����. = �?

7���5� �Ï = �( = �����., �ℎ�� �è� = 0 , �Qè

�Q = 0 , ��"�. �� ��!���� ��J

0 + 0 + Ò �Ï = �? , Ò�( = � , ∴ �( = e�Ü

3- 2nd

Order non homog. Linear equn

Page 25: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 25

�� 1: −���!� �Q�Q + 2 �

� − 3� = 6

���J: − � = �e + �Ï

����� ���!� ��� �e �ℎ��ℎ �� ��5 ��5������ ���J

�Q×�Q + 2 �×

� − 3�e = 0 , �,( + 2, − 3#�e = 0 , �5( + 25 − 3# = 0

�5 + 3#�5 − 1# = 0 , �� �? = −3 , & �( = 1 �ℎ��ℎ �� ���� 1: �? ≠ �(

�� �e = 7 �KM + ± �

Y�� ���!� ��� ��������� �������� G�� �Ï = � , �è

� = 0 , �Qè�Q = 0 , ��"�. �� 1ÌÍ��J: 0 + 0 − 3� = 6 →

� = −2 , ∴ �Ï = −2

∴ �ℎ� 6������ �������� ���� "� � = �e + �Ï, �� � = 7 �KM + ±� − 2

Case (b): If ç��# = okq�rq|igk = gq + gf� + gz�z + − − − + gr�r

7���5� �Ï = à[ + à?� + à(�( + − − − − − + àJ�J

�ℎ�� ���� �è� , �Qè

�Q , ��� ��"�. �� 5��� ��J �� ���� à[ , à?, à(, … . . àJ

�� 2: −���!� �'' − 4 �' + 4 � = 4� + 8�M

1ÌÍ ���!� ��� ��5 ��5������ ���J ∶ �e

�Q×�Q − 4 �×

� + 4 �e = 0 �,( − 4, + 4#�e = 0 , �5( − 45 + 4# = 0 , �5 − 2#�5 − 2# = 0 ∴ �? = �( = 2 , ���� �"# , ∴ �e = �7� + ±#�(

Page 26: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 26

2J� ���!� ��� ��������� ���J , �Ï

7���5� �Ï = à[ + à?� + à(�( + àM�M

�è� = à? + 2à(� + 3àM�(

�Qè�Q = 2à( + 6àM� , ��"�. �� 5��� ��J 2à( + 6àM� − 4�à? + 2à(� + 3àM�(# + 4�à[ + à?� + à(�( + àM�M# = 4� +8�M

2à( + 6àM� − 4à? − 8à(� − 12àM�( + 4à[ + 4à?� + 4à(�( + 4àM�M = 4� +8�M

����������� �� �: �6àM − 8à( + 4à?# = 4

����������� �� �(: �4à( − 12àM# = 0

����������� �� �M: 4àM = 8 , →→ àM = ÅO = 2

��� 4à( = 24 , →→ à( = (OO = 6, ��� 12 − 48 + 4à? = 4 →→ à? = OI

O 10

��� à[ , �� ℎ�!�: 2à( − 4à? + 4à[ = 0, �� à[ = (ÅO = 7

∴ à[ = 7 , à? = 10 , à( = 6 , ��� àM = 2 �Ï = 7 + 10 � + 6 �( + 2 �M , ��� .. . � = �Ï + �e , .. . � = �7 � + ±#�( + 7 + 10 � + 6 �( + 2 �M. Case (c) If ç��# = � lh� , �&�: �����

7���5� �Ï = à �R , �� �è� = à � �R , �Qè

Q = à �( �R

�ℎ�� ��"������� �� �ℎ� 5��� ��J �� 6�� à.

Page 27: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 27

�� 3: − ���!� �'' − 6�' + 9� = 8 �

���J: 1ÌÍ ���!� ��� ��5 ��5������ �������� �e

�,( − 6, + 9#�e = 0 , �5( − 65 + 9# = 0 , �5 − 3#�5 − 3# = 0

∴ �? = �( = 3 ���� �"# , �e = �7� + ±#�M

2J� ���!� ��� ��������� �������� �Ï

7���5� �Ï = à � , �è� = à � , ��� �Qè

�Q = à � �� �ℎ� 5��� �������� ∶ à� − 6 à � + 9 à � = 8�

�à − 6à + 9 à#� = 8 �

4à = 8 →→ à = 2 , ∴ �Ï = 2 � , ∴ � = �7 � + ±#�M + 2 �

Y���: $� �ℎ� ����� �� �ℎ� ��5 ; �5������ �������� ��5� �� �ℎ� ���� �� �Ï �ℎ��

�ℎ��6� �ℎ� ����5���� �� �Ï �� à � �R ��� �� �� ���� ���� �ℎ�� �ℎ��6� �� à �(�R ��� ���!� . �� 4: − ���!� �'' − 6 �' + 9 � = 8 �M ���J: − �,( − 6, + 9#�e = 0 , �5( − 65 + 9# = 0 , �5 − 3#�5 − 3# = 0

∴ �? = �( = 3 →→ ���� �1#, ��� �e = �7 � + ±#�M

Y�� �Ï = à �M, "�� �� �� ����� �ℎ�� ����5� �Ï = à � �M ,"�� ���� ����� , �� �� ����5� �Ï = à �(�M, ��� ���!� �è� = 3à �(�M + 2 à � �M , �Qè

�Q = 6à��M + 9 à �(�M + 2à �M + 6à� �M

Y�� ��"��. �� 5��� ��J ��� ������ �ℎ� ������������

12 à � �M − 9 à�(�M + 2à �M − 12 à � �M + 9 à �(�M = 8 �M

2 à �M = 8 �M →→ 2à = 8 , ∴ à = 4

∴ �Ï = 4�(�M , ∴ � = �7� + ±#�M + 4 �( �M

Page 28: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 28

Case (d) : If ç��# = é �qm ê� qh é mir ê� 7���5� �Ï = à cos �� + á sin �� , �ℎ�� ���� �è

� & �Qè�Q , ��� ��"��. ��

�ℎ� 5��� �������� �� 6�� à, ��� á. �� 5: − �'' + 2�' + 10� = 100 cos 4� ���J: − 1ÌÍ ���!� ��� ��5 ��5������ �������� �'' + 2�' + 10� = 0 , �,( + 2, + 10#�e = 0 , �5( + 25 + 10# = 0

5 = K( ∓√OKOI( = −1 ∓ 3� , �? = −1 + 3� , �( = −1 − 3�

�� à = −1 , & á = 3 , ��� �e = �K�7 cos 3� + ± sin 3�#

Y�� ���!� ��� ��������� �������� , 7���5� �Ï = à cos 4� + á sin 4�

�Ï' = −4à sin 4� + 4á cos 4� , �Ï'' = −16 à cos 4� − 16 á sin 4�

−16 à cos 4� − 16 á sin 4� − 8à sin 4� + 8á cos 4� + 10 à cos 4� +10 á sin 4� = 100 cos 4� −6 à + 8á = 100, −6á − 8à = 0, →→ á = − Å

µ à

−6 à − µOµ à = 100 , →→ − ?II

µ à = 100 , ∴ à = −6 , á = 8

∴ �Ï = −6 cos 4� + 8 sin 4�

��� � = �K�7 cos 3� + ± sin 3�# − 6 cos 4� + 8 sin 4�

Y���: $� Õ��# = à �J�R cos �� , �� Õ��# = à �J�R sin ��

7���5� ∶ �Ï = �7[�J + − − − + 7JK?� + 7J#�R cos �� + ��7[�J + − −− + 7JK?� + 7J#�R sin ��

Page 29: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 29

=��!� �ℎ� ��������6 ������������ ��J�: −

1# �'' + 2�' + � = 2 �K, 2# 5�'' + 12�' + 20� = 120 sin 2�

3# 5�'' + 6�' + 2� = �( + 6�

4# �'' − 4�' − 5� = � + 4

5# �'' − � = � + �(

6# �'' − �' − 6� = �K − 7 cos �

7# �'' − 2�' + � = �M� .

$� �� �������� �� �ℎ� ���5: − �J�J + �JK?�JK? + − − − − +�(�''' + �?�'' + �[�' = 0

$� �� ℎ�6ℎ�� ����� ℎ�5�6����� �������� ��� ��'������ ��� �?, �(, �M, . . . �J

�ì = �?�R� + �( �RQ + �M �RU + − − − − +�J �RP

$� �? = �( = �M = − − − − −= �J

�ì = �RPÚ�? + �(� + �M�( + − − − −]

$� �� �������� ∶ �J�J + �JK?�JK? + − − − − +�(�''' + �?�'' = ���#

$� �� ℎ�6ℎ�� ����� ���ℎ�5�6����� �������� ���!�� �� �ℎ� ��5� �� �� ������

"����� ��� 2J� ����� ���ℎ�5�6����� ��������

�� 1: − ���!� �ℎ� ¶. ,. � �''' − 3�'' + 2�' = 0

���J : �U�U − 3 �Q

�Q + 2 �� = 0 , �,M − 3 ,( + 2,#� = 0

��M − 3 �( + 2�# = 0 , ���( − 3 � + 2# = 0 , ��� − 2#�� − 1# = 0

.. . �? = 0 , �( = 2 , �M = 1, .. . �ì = �?�I + �(�( + �M�

{i}�lh íh�lh �isslhlrpigk ��~gpiqr

H.W 7

Page 30: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 30

�ì = �? + �(�( + �M�

�� 2: − ���!� �ℎ� ¶. ,. � ∶ �'''' + 2 �''' − 3 �'' = �( + 3�( + 4 sin �

���J: 1� ���� �e ��� �'''' + 2�''' − 3�'' = 0

�O + 2�M − 3�( = 0 , �(��( + 2� − 3# = 0 , �(�� + 3#�� − 1# = 0

.. . �? = �( = 0 , �M = −3 , �O = 1

�e = �IÚ�?� + �(] + �M�KM + �O� , �e = �?� + �( + �M�KM + �O�

1� ���� �Ï , ��� �Ï = �Ï? + �Ï( + �ÏM

�Ï? = 7�O + ±�M + ��(, "������ �? = �( = 0 & �� ℎ�!� �(, ��� �?�

�Ï( = , �( , �ÏM = � cos � + N sin �

�Ï = 7�O + ±�M + ��( + , �( + � cos � + N sin �

�Ï' = 47�M + 3±�( + 2·� + 2,�( − � sin � + N cos �

�Ï'' = 127�( + 6±� + 2· + 4,�( − � cos � − N sin �

�Ï''' = 247� + 6± + 8,�( + � sin � − N cos �

�Ï'''' = 247 + 16,�( + � cos � + N sin �

247 + 16,�( + � cos � + N sin � + 487� + 12± + 16,�( + 2� sin � −2N cos � − 36 7 �( − 18 ± � − 6· − 12 , �( + 3 � cos � + 3 N sin � = �( +3 �( + 4 sin �

20 , �( = 3 �( →→ , = M(I , −36 7 �( = �( →→ 7 = − ?

�48 7 − 18 ±#� = 0 →→ ± = − (î(

24 7 − 6 · = 0 →→ · = −9

�4N + 2�# sin � = 4 sin � , �4� − 2N# cos � = 0

Page 31: Chapter Two year/Mathe… · Chapter two(Ordinary Differential Equations) Page 1 Definition: Any equation involving one derivative or more in the unknown function (the dependent variable)

Chapter two(Ordinary Differential Equations) Page 31

4� = 2N , →→ N = 2 � , 4N + 2� = 4 →→ 8� + 2� = 4 →→ � = 25 ,

N = 45

�Ï = − ?Mµ �O − (î

( �M − 9�( + M(I �( + (

¡ cos � + O¡ sin �

.. . � = �?� + �( + �M�KM + �O� − ?Mµ �O − (î

( �M − 9�( + M(I �( + (

¡ cos � +O¡ sin �

=��!� �ℎ� ��������6 ,. �: −

1# �(�'' − � �' − 3� = �( ln �. 2# �'' − 2�' + � = �M� . 3# �'' − �' − 6� = �K − 7 cos �

4# 5�'' + 12�' + 20� = 120 sin 2�. 5# �'' − 4�' + 8� = �( sin 2�. 6# �'' + 4� = 4 sin 2�. 7# �'' − 2�' + 2� = � cos �. 8# 2�(� �Q

�Q + �( = �( u��v(.

9# �'' = 1 + �'(. 10# � �'' + �� + 1#�'( = 0

11# � �'' + �' = 0. 12# �(�'' + � �' = 1. 13# �(�'' + 3��' − 3� = 0. 14# �(�'' + 3��' − 3� = 1. 15# �(�'' + ��' − � = 0. 16# �(�'' − 2��' + 2� = �M cos �. 17# �''' − 4�'' = � + �O.

H.W 8