tuesday february 15, 2011 - penn mathryblair/math240/papers/lec2...math 240: linear diﬀerential...

Math 240: Linear Differential Equations

Ryan Blair

University of Pennsylvania

Tuesday February 15, 2011

Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 1 / 12

Outline

1 Review

2 Today’s Goals

3 Undetermined Coefficients


Review

Review for Last Time

1 Construct general solutions to homogeneous andnonhomogeneous linear D.E.s

2 Use auxiliary equations to solve constant coefficient linearhomogeneous D.E.s


Review

Auxiliary Equations

Given a linear homogeneous constant-coefficient differentialequation

andny

dxn + an−1dn−1y

dxn−1 + ...a1dy

dx+ a0y = 0,

the Auxiliary Equation is

anmn + an−1m

n−1 + ...a1m + a0 = 0.


Review

Auxiliary Equations

Given a linear homogeneous constant-coefficient differentialequation

andny

dxn + an−1dn−1y

dxn−1 + ...a1dy

dx+ a0y = 0,

the Auxiliary Equation is

anmn + an−1m

n−1 + ...a1m + a0 = 0.

The Auxiliary Equation determines the general solution.


Review

General Solutions to Nonhomogeneous Linear D.E.s

Theorem

Let yp be any particular solution of the nonhomogeneous linear

nth-order differential equation on an interval I . Let y1, y2, ..., yn be a

fundamental set of solutions to the associated homogeneous

differential equation. Then the general solution to the

nonhomogeneous equation on the interval is

y = c1y1(x) + c2y2(x) + ... + cnyn(x) + yp

where the ci are arbitrary constants.


Review

General Solution from the Auxiliary Equation

1 If m is a root of the auxiliary equation of multiplicity k thenemx , xemx , x2emx , ... , xk−1emx are linearly independentsolutions.

2 If (α + iβ) and (α + iβ) are a roots of the auxiliary equation ofmultiplicity k theneαxcos(βx), xeαxcos(βx), ... , xk−1eαxcos(βx) andeαxsin(βx), xeαxsin(βx), ... , xk−1eαxsin(βx) are linearlyindependent solutions.


Today’s Goals

Today’s Goals

1 Learn how to solve nonhomogeneous linear differential equationsusing the method of Undetermined Coefficients.


Undetermined Coefficients

The Method of Undetermined Coefficients

Given a nonhomogeneous differential equation

any(n) + an−1y

(n−1) + ...a1y′ + a0y = g(x)

where an, an−1, ..., a0 are constants.





any(n) + an−1y

(n−1) + ...a1y′ + a0y = g(x)


1 Step 1: Solve the associated homogeneous equation.





any(n) + an−1y

(n−1) + ...a1y′ + a0y = g(x)


1 Step 1: Solve the associated homogeneous equation.2 Step 2: Find a particular solution by analyzing g(x) and making

an educated guess.





any(n) + an−1y

(n−1) + ...a1y′ + a0y = g(x)


1 Step 1: Solve the associated homogeneous equation.2 Step 2: Find a particular solution by analyzing g(x) and making

an educated guess.

3 Step 3: Add the homogeneous solution and the particularsolution together to get the general solution.



Guessing Particular Solutions

g(x) Guessconstant




g(x) Guessconstant A





3x2− 2





3x2− 2 Ax2 + Bx + C





3x2− 2 Ax2 + Bx + C

Polynomial of degree n





3x2− 2 Ax2 + Bx + C

Polynomial of degree n Anxn + An−1x

n−1 + ... + A0





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x)





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x) Acos(4x) + Bsin(4x)





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx)





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)e4x





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)e4x Ae4x





3x2− 2 Ax2 + Bx + C


n−1 + ... + A0

cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)e4x Ae4x

x2e5x (Ax2 + Bx + C )e5x

e2xcos(4x) Ae2x sin(4x) + Be2xcos(4x)3xsin(5x) (Ax + B)sin(5x) + (Cx + D)cos(5x)xe2xcos(3x) (Ax + B)e2xsin(3x) + (Cx + D)e2xcos(3x)



The Guessing Rule

The form of yp is a linear combination of all linearly independentfunctions that are generated by repeated differentiation of g(x).



A Problem

Solve y ′′− 5y ′ + 4y = 8ex using undetermined coefficients.



The solution

When the natural guess for a particular solution duplicates ahomogeneous solution, multiply the guess by xn, where n is thesmallest positive integer that eliminates the duplication.


tuesday february 15, 2011 - penn mathryblair/math240/papers/lec2...math 240: linear diﬀerential...

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