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Suppose $y_1(x)= e^{-2x} + xe^{-x}$ , $y_2(x)= xe^{-2x} + xe^{-x}$ , $y_3(x)= e^{-2x} - xe^{-2x} + xe^{-x}$ are three special solutions to the differential equation,

$y'' + a_1y' + a_2y = F(x)$ , $a_1$ and $a_2$ are constants.

Find $a_1, a_2$ and $F(x$).

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Hint: $y_1'' + a_1 y_1' + a_2 y_1 =y_2'' + a_1 y_2' + a_2 y_2 $ and $y_2'' + a_1 y_2' + a_2 y_2 = y_3'' + a_1 y_3' + a_2 y_3$ are two equations in two unknowns $a_1$, $a_2$. Solve. Then look at $y_1'' + a_1 y_1' + a_2 y_1$ to find $F(x)$.

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  • $\begingroup$ Is there another (shorter) way to solve this? $\endgroup$
    – Sam
    Sep 8, 2014 at 3:40
  • $\begingroup$ @Sam. This is the simplest and most rational way to solve your problem. $\endgroup$ Sep 8, 2014 at 5:34

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