Question

Find the differential equations of the family of curves $y=e^x$ $(A\cos x+B\sin x)$ where A and B are arbitrary constants.

• A. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+2y=0$
• B. $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$
• C. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+2y=0$
• D. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-2y=0$

Answer 1

The correct option is B $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$

$y=e^x$ $(A\cos x+B\sin x)......\left(1\right)$

$y_1=e^x(A\cos x+B\sin x)+e^x(-A\sin x+B\cos x)$

or $y_1-y=e^x(-A\sin x+B\cos x)\cdots \left(2\right)$

Differentiating again w.r.t.x, we get
$y_2-y_1=e^x(-A\sin x+B\cos x)+e^x(-A\cos x-B\sin x)$

or $y_2-y_1=(y_1-y)-y,$ by (1)and (2)

or $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$, where $y_1=\frac{dy}{dx},y_2=\frac{d^{2}y}{d{x}^2}$