Question

Obtain a differential equation by eliminating the arbitrary constants $A,B$ from the following equation.

$y=A\cos t-B\sin t$.

Answer 1

Given,

$y=A\cos t-B\sin t$......(1).

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

$\frac{dy}{dt}=-A\cos t-B\sin t$

Again differentiating both sides with respect to $t$ we get,

$\frac{d^{2}y}{d{t}^2}=-A\cos t+B\sin t$

or, $\frac{d^{2}y}{d{t}^2}=-y$ [ Using (1)]

or, $\frac{d^{2}y}{d{t}^2}+y=0$

This is the required differential equation.