Question

Form a differential equation by eliminating the arbitrary constants $a,b$ from the equation:

$y=a{e}^{kx}+b{e}^{-kx}$.

Answer 1

Given,

$y=a{e}^{kx}+b{e}^{-kx}$.......(1).

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}=k(a{e}^{kx}-b{e}^{-kx})$

Again differentiating both sides we've,

$\frac{d^{2}y}{d{x}^2}=k^2(a{e}^{kx}+b{e}^{-kx})$

or, $\frac{d^{2}y}{d{x}^2}=k^{2}y$. [ Using (1)]

This is the required differential equation.