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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
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 2 / 12
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
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 3 / 12
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 4 / 12
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 4 / 12
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 5 / 12
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 6 / 12
Today’s Goals
Today’s Goals
1 Learn how to solve nonhomogeneous linear differential equationsusing the method of Undetermined Coefficients.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 7 / 12
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 8 / 12
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.
1 Step 1: Solve the associated homogeneous equation.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 8 / 12
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.
1 Step 1: Solve the associated homogeneous equation.2 Step 2: Find a particular solution by analyzing g(x) and making
an educated guess.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 8 / 12
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.
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 8 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
cos(4x)
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
cos(4x) Acos(4x) + Bsin(4x)
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx)
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)e4x
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
n−1 + ... + A0
cos(4x) Acos(4x) + Bsin(4x)Acos(nx) + Bsin(nx) Acos(nx) + Bsin(nx)e4x Ae4x
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
Guessing Particular Solutions
g(x) Guessconstant A
3x2− 2 Ax2 + Bx + C
Polynomial of degree n Anxn + An−1x
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)
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 9 / 12
Undetermined Coefficients
The Guessing Rule
The form of yp is a linear combination of all linearly independentfunctions that are generated by repeated differentiation of g(x).
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 10 / 12
Undetermined Coefficients
A Problem
Solve y ′′− 5y ′ + 4y = 8ex using undetermined coefficients.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 11 / 12
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.
Ryan Blair (U Penn) Math 240: Linear Differential Equations Tuesday February 15, 2011 12 / 12
