Nonhomogeneous Equations; Method of
Undetermined Coefficients
MATH 365 Ordinary Differential Equations
J. Robert Buchanan
Department of Mathematics
Summer 2008
J. Robert Buchanan Nonhomogeneous Equations; Method of Undetermined Coefficients
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Nonhomogeneous Equations, Method of Undetermined Coefficients | MATH 365, Assignments of Differential Equations
Material Type: Assignment; Professor: Buchanan; Class: Ordinary Differential Equation; Subject: Mathematics; University: Millersville University of Pennsylvania; Term: Unknown 2008;
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Nonhomogeneous Equations; Method of
Undetermined Coefficients
MATH 365 Ordinary Differential Equations
J. Robert Buchanan
Department of Mathematics
Summer 2008
Background
Recently we have concentrated on finding solutions to second order linear, homogeneous ODEs of the form
L [ y ] = y ′′^ + p ( t ) y ′^ + q ( t ) y = 0.
Today we return to the case of nonhomogeneous second order linear ODEs of the form
L [ y ] = y ′′^ + p ( t ) y ′^ + q ( t ) y = g ( t ),
where p , q , and g are continuous functions on an open interval I.
Nonhomogeneous vs. Homogeneous Solutions
Theorem If Y 1 and Y 2 are two solutions of a nonhomogeneous second order linear ODE, then their difference Y 1 − Y 2 is a solution to the corresponding homogeneous second order linear ODE. If, in addition, y 1 and y 2 are a fundamental set of solutions to the homogeneous second order linear ODE, then
Y 1 ( t ) − Y 2 ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t )
where c 1 and c 2 are constants.
Proof.
General Solution
Theorem The general solution of a nonhomogeneous second order linear ODE can be written as
y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) + Y ( t )
where y 1 and y 2 are a fundamental set of solutions to the corresponding homogeneous second order linear ODE, c 1 and c 2 are arbitrary constants and Y is some specific solution to the nonhomogeneous second order linear ODE.
Method of Undetermined Coefficients
The method of undetermined coefficients can be used to find the particular solution when g ( t ) contains polynomials, sines and cosines (but not the other trigonometric functions), exponential functions, or sums and products of these types of functions.
Example
Example Find the general solution to the nonhomogeneous second order linear ODE: y ′′^ + 4 y ′^ + 9 y = t^2.
Example
Example Find the general solution to the nonhomogeneous second order linear ODE: y ′′^ − 4 y = 4 e^2 t^.
Example
Example Find the solution to the nonhomogeneous second order linear IVP:
y ′′^ + 16 y = 5 sin t y ( 0 ) = 0 y ′( 0 ) = 0
Example
Example Find the general solution to the nonhomogeneous second order linear ODE: 4 y ′′^ + y = t^2 + 2 cos 3 t.
Example
Example Find the general solution to the nonhomogeneous second order linear ODE: y ′′^ − 3 y ′^ − 4 y = − 8 et^ sin t.
Example
Example Find the general solution to the nonhomogeneous second order linear ODE:
y ′′^ − 5 y ′^ + 6 y = e^2 t^ ( t + 2 ) cos t.
Summary
ay ′′^ + by ′^ + cy = g ( t )
g ( t ) Y ( t ) Pn ( t ) = a 0 + a 1 t + · · · + antn^ tk^ ( A 0 + A 1 t + · · · + Antn ) Pn ( t ) eat^ tk^ ( A 0 + A 1 t + · · · + Antn ) eat
Pn ( t ) eat
sin bt cos bt
tk^ ( A 0 + A 1 t + · · · + Antn ) eat^ cos bt
- tk^ ( B 0 + B 1 t + · · · + Bntn ) eat^ sin bt