Question

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

Answer 1

We have $y=c_1{e}^{2x}+c_2{e}^{-2x}$

Differentiate with respect to $x$ we get

$\frac{dy}{dx}=2c_1{e}^{2x}-2c_2{e}^{-2x}$

Again differentiate with respect to $x$ we get

$\frac{d^{2}y}{d{x}^2}=4c_1{e}^{2x}+4c_2{e}^{-2x}$

$=4(c_1{e}^{2x}+c_2{e}^{-2x})$

$=4y$

$\therefore \frac{d^{2}y}{d{x}^2}-4y=0$