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Please help with the following problem:

Let $ϕ$ be a solution of the equation $y''+a_1y'+a_2y=0$, where $a_1,a_2$ are constants. If $ψ(x) = e^{(\frac{a_1}{2})x}ϕ(x)$, show that $ψ$ satisfies an equation $y''+ky=0$, where $k$ is some constant. Compute $k$.

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    Do you mind sharing your own thoughts on the problem? – 2017-02-19
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    what have you done already? – 2017-02-19

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Hint: $$ψ(x) = e^{\frac{a_1}{2}x}ϕ(x)$$ $$ψ'(x) = \frac{a_1}{2}e^{\frac{a_1}{2}x}ϕ(x)+e^{\frac{a_1}{2}x}ϕ'(x)$$ and take second derivate $ψ''(x)$ then substitute in $\dfrac{y''}{y}$.

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    Substitute in to $y′′+ky=0$? – 2017-02-19
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    You can substitute in $y''+ky$ and show it is zero for a $k$, but my offer is better, because you don't know what's $k$. – 2017-02-19