Prove that
$y = 2 e^{x} + 3 e^{2} x$ is not g.s. of differential equation $y^{2} - 3 y + 2 y = 0$

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Solution

Given,

$y = (2 e^{x} + 3 e^{2 x})$

$y^{2} - 3 y + 2 y = 0$

$y^{2} - y = 0$

$((2 e^{x} + 3 e^{2 x}))^{2} - (2 e^{x} + 3 e^{2 x}) = 0$

$4 e^{2 x} + 12 e^{3 x} + 9 e^{4 x} - 2 e^{x} - 3 e^{2 x} = 0$

$- 2 e^{x} + e^{2 x} + 12 e^{3 x} + 9 e^{4 x} \neq 0$

Hence proved.

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