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MATH150T6

# MATH150T6 - MATH 150 Inhomogeneous second order linear ODE...

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MATH 150 Inhomogeneous second order linear ODE Cheung Man Wai Tutorial 6 Now consider the inhomogeneous ODE ¨ x + p ( t ) ˙ x + q ( t ) x = g ( t ) , - - - ( * ) with initial conditions x ( t 0 ) = x 0 and ˙ x ( t 0 ) = u 0 . The standard procedure for finding the solution of (*) can be described as follows. Step 1 Find the general solution of the homogeneous equation ¨ x + p ( t ) ˙ x + q ( t ) = 0 . - - - ( ** ) Denote the homogeneous solution by x h ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) , where x 1 and x 2 are linearly independent solutions of (**), and c 1 , c 2 are con- stants to be chosen later. Step 2 Find any particular solution x p ( t ) of (*), applying the method of unde- termined coefficients . Step 3 The general solution of (*) is given by x ( t ) = x h ( t ) + x p ( t ) , where c 1 and c 2 should be determined by the initial conditions. In the examples below we focus our effort on finding particular solutions, for the purpose of revealing the spirit of undetermined coefficients as much as possible. Example 1 Compute a particular solution to the equation y + 2 y - 3 y = 5 sin 3 t.

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