Question

If $y=x^{2} e^{x}$, show that $\frac{d^{2}y }{dx^{2}} -\frac{dy}{dx}-2(x+1)e^{x}=0$

Answer 1

$\underline{\bold{Solution:-}}$

$y = {x}^{2} {e}^{x} \: \: .....(1)\\ \\ on \: differentiating \: both \: sides \\ \\ \frac{dy}{dx} = \frac{d}{dy} ( {x}^{2} {e}^{x} ) \\ \\ by \: using \: product \: rule \\ = {x}^{2} \frac{d}{dy} {e}^{x} + {e}^{x} \frac{d}{dy} {x}^{2} \\ \\ = {x}^{2} {e}^{x} + {e}^{ x } (2x) \\ \\ = {e}^{x} x(x + 2)$

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