7/26/2019 Differential Equation Semester 2 Exercise

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[ENGINEERING MATHEMATICS 2: LGB 10403]

Chapter 3: Differential Equation

3.2 First Order Differential Equation.

3.2.1 Solving First Order Differential Equation.

3.2.1.3 Method 3: Homogeneous Equation.

1. An equation of the form

Qdx

dyP =

where P and are fun!tions of "oth # and $ of the samedegree% is said to "e homogeneous in $ and #.

2. Formula& Su"stitute

3. E#am'le&

+

=

++=

22

2

22

2(%)

3(%)

yx

yxyxf

yxyxyxf

3. E#am'le&

Solve

xy

yx

dx

dy 22 +=

Solution:

Step1: Form the equation of

P

Q

dy

dx=

Step 2: se

vxy=

and differentiate !

Step3: Su"stitute

vxy=

into

xy

yx

dx

dy 22 +=

7/26/2019 Differential Equation Semester 2 Exercise

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[ENGINEERING MATHEMATICS 2: LGB 10403]

Step #: Sol$e differentiation equation

Step %: &epla'e (ith

vxy=

*. E#er!ise&

)a( Solve differential equation

dx

dyxxy =

)"( Determine the 'arti!ular solution of the equation

y

yx

dx

dyx

22+

=

given that the "oundar$

!onditions that $+* and #+1.

)!( Solve

dx

dyyx () 22 +

)d( Determine the general solution of

,=+dx

dyxyx

)e( Determine the 'arti!ular solution of the differential equation

xydxdyyx =+ () 22

given that#+1 and $+1.

7/26/2019 Differential Equation Semester 2 Exercise

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[ENGINEERING MATHEMATICS 2: LGB 10403]

)f( Determine the 'arti!ular solution of the differential equation

1(2

2) =

+

dx

dy

xy

xy

given that $+3when #+2.

3.2 First Order Differential Equation.

3.2.1 Solving First Order Differential Equation.

3.2.1.# Method #: )inear Differential Equation.

1. An equation of the form

QPydx

dy=+

where P and are fun!tions of # onl$ is !alled a lineardifferential equation sin!e $ and its differentiation are of the first degree.

2. Formula& -se

3. E#am'le&

Show that the solution of the equation

x

y

dx

dy=+1

is given "$

x

xy

2

3 2=

given #+1 and$+1

Solution:

Step1: Form the equation into

QPydydx =+

form

Step 2: Determine

Pdx

Step3: Determine the integrating fa'tor

Pdxe

7/26/2019 Differential Equation Semester 2 Exercise

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[ENGINEERING MATHEMATICS 2: LGB 10403]

Step #: Su"stitute into

Qdxeye PdxPdx

=

Step %: Sol$e the general solution and parti'ular solution

*. E#er!ise&

)a( Determine the 'arti!ular solution of the equation

,=+ yxdx

dy

given that the "oundar$!onditions that #+, when $+2.

)"( Determine the differential equation

tanse! yd

dy+=

given that the "oundar$ !onditions

that $+1 when

,=.

)!( i Solve the general solution of the equation

1(1)(1)3(2) =++

yxx

dxdyx

ii /iven that the "oundar$ !onditions that $+0 when #+1% determine the 'arti!ular solution ofthe equation given in )i(.