The D.E whose solution is $y = C_{1} e^{2 x} + C_{2} e^{- x / 2}$ is:

2 y_{2} - 3 y_{1} - 2 y = 0

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3 y_{2} - 2 y_{1} - y = 0

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y_{2} - 3 y_{1} + y = 0

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y_{2} - 3 y_{1} - y = 0

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Solution

The correct option is A $2 y_{2} - 3 y_{1} - 2 y = 0$
$y = C_{1} e^{2 x} + C_{2} e^{- x / 2}$
$\Rightarrow y_{1} = 2 C_{1} e^{2 x} - \frac{C_{2}}{2} e^{- x / 2}$
$\Rightarrow 2 y_{1} = 4 C_{1} e^{2 x} - C_{2} e^{- x / 2}$
$\Rightarrow 2 y_{1} + y = 5 C_{1} e^{2 x}$
$\Rightarrow 2 y_{1} + y = 5 C_{1} e^{2 x}$
$\Rightarrow 2 y_{2} + y_{1} = 10 e^{2 x}$
$= 4 y_{1} + 2 y$
$\Rightarrow 2 y_{2} - 3 y_{1} - 2 y = 0$

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