Find the differential equation whose solution represents the family: $y = a e^{3 x} + b e^{x}$

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Solution

$\begin{matrix} y = a e^{3 x} + b e^{x} \frac{d y}{d x} = 3 a e^{3 x} + b e^{x} \frac{d^{2} y}{d x^{2}} = 9 a e^{3 x} + b e^{x} \frac{d y}{d x} - y = 2 a e^{3 x} \frac{1}{2} [\frac{d y}{d x} - y] = a e^{3 x} y - \frac{1}{2} [\frac{d y}{d x} - y] = b e^{x} \frac{3 y}{2} - \frac{1}{2} \frac{d y}{d x} = b e^{x} \frac{d^{2} y}{d x^{2}} = \frac{9}{2} [\frac{d y}{d x} - y] + \frac{3 y}{2} - \frac{1}{2} \frac{d y}{d x} \frac{d^{2} y}{d x^{2}} = 4 \frac{d y}{d x} - 3 y \frac{d^{2} y}{d x^{2}} - 4 \frac{d y}{d x} + 3 y = 0 \end{matrix}$

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Find the differential equation whose solution represents the family: $y = a e^{3 x} + b e^{x}$