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Find the general solution of y′′ – y = x

Find the general solution of the second-order differential equation

    \[ y'' - y = x. \]

If the solution is not valid everywhere, describe the interval on which it is valid.


The general solution of the homogeneous equation

    \[ y'' - y = 0 \]

is given by Theorem 8.7 with a = 0 and b = -1. This gives us d = a^2 - 4b = 4; hence, k = \frac{1}{2} \sqrt{d} = 1. Thus,

    \begin{align*}  y &= e^{-\frac{ax}{2}} (c_1 e^x + c_2 e^{-x} ) \\  &= c_1 e^x + c_2 e^{-x}. \end{align*}

To find a particular solution of y'' - y = x let y_1 = Ax + B. Then, y_1'' = 0 implies

    \[ -Ax-B = x \quad \implies \quad A = -1, \ \ B = 0 \quad \implies \quad y_1 = -x. \]

By Theorem 8.8 (page 330 of Apostol) the general solution of the given differential equation is then

    \[ y = c_1 e^x + c_2e^{-x} -x. \]

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