Show that $y = a e^{2 x} + b e^{- x}$ is a solution of the differential equation $\frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} - 2 y = 0$ .

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Solution

given differential equation $\frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} - 2 y = 0.... (1)$
$y = a e^{2 x} + b e^{- x} . . . . (2)$
differentiating on both sides
$2 a e^{2 x} - b e^{- x}$
Again differentiating on both sides
$4 a e^{2 x} + b e^{- x}$
substituting in (1) $(4 a e^{2 x} + b e^{- x}) - (2 a e^{2 x} - b e^{- x}) - 2 (a e^{2 x} + b e^{- x}) = 0$
So, (2) satisfies eqation (1)
$y = a e^{2 x} + b e^{- x}$ is a solution of (1)

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