Obtain the differential equation by eliminating the arbitrary con… - Vidyadip

Question

Obtain the differential equation by eliminating the arbitrary constants from the following equation:
$
y=c_1 e^{2 x}+c_2 e^{-2 x}
$

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Answer

$y=c_1 e^{2 x}+c_2 e^{-2 x}$
differentiate w.r.t. x.
$\frac{d y}{d x}=2 c_1 e^{2 x}-2 c_2 e^{-2 x}$
Again diff. w.r.t. x.
$\begin{aligned} & \frac{d^2 y}{d x^2}=4 c_1 e^{2 x}+4 c_2 e^{-2 x} \\ & =4\left(c_1 e^{2 x}+c_2 e^{-2 x}\right) \\ & =4 y \\ & \therefore \frac{d^2 y}{d x^2}-4 y=0\end{aligned}$

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