Question

Obtain a differential equation by eliminating the arbitrary constants $a$ and $b$ from the equation $y=a\cos nt+b\sin nt$.

Answer 1

Given,

$y=a\cos nt+b\sin nt$......(1).

Differentiate both sides with respect to $t$ we get,

$\frac{dy}{dt}=n(-a\sin nt+b\cos nt)$

Again differentiating with respect to $t$ we get,

or, $\frac{d^{2}y}{d{t}^2}=-n^2(a\cos nt+b\sin nt)$

or, $\frac{d^{2}y}{d{t}^2}=-n^{2}y$

or, $\frac{d^{2}y}{d{t}^2}+n^{2}y=0$ [ Using (1)].

This is the required differential equation.