Question

If $y=e^{ax}\cos bx$, then prove that

$\frac{d^{2}y}{d{x}^2}-2a\frac{dy}{dx}+(a^2+b^2)y=0$

Answer 1

$y=e^{ax}\cos bx$ ... (i)

Differentiating (i) w.r.t. $x,$ we get,

$\frac{dy}{dx}=e^{ax}a\cos bx+e^{ax}(-\sin bx)b$

$\Rightarrow \frac{dy}{dx}=ay-b{e}^{ax}\sin bx$

$\Rightarrow \frac{dy}{dx}-ay=-b{e}^{ax}\sin bx$ ... (ii)

Again differentiating w.r.t. $x,$ we get,

$\Rightarrow \frac{d^{2}y}{d{x}^2}-a\frac{dy}{dx}=-b{e}^{ax}a\sin bx-b{e}^{ax}b\cos bx$

$\Rightarrow \frac{d^{2}y}{d{x}^2}-a\frac{dy}{dx}=(-b{e}^{ax}\sin bx)a-b^2\left(e^{ax}\cos bx\right)$

Using (i) and (ii), we have

$\Rightarrow \frac{d^{2}y}{d{x}^2}-a\frac{dy}{dx}=(\frac{dy}{dx}-ay)a-b^{2}y$

$\Rightarrow \frac{d^{2}y}{d{x}^2}-2a\frac{dy}{dx}+(a^2+b^2)y=0$