Question

The differential equation of the family of curves $y=e^x(A\cos x+B\sin x)$, where $A$ and $B$ are arbitrary constants, is

• 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. $\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2+y=0$
• D. $\frac{d^{2}y}{d{x}^2}-7\frac{dy}{dx}+2y=0$

Answer 1

Answer: (A)

$y=e^x(A\cos x+B\sin x)$
$\frac{dy}{dx}=e^x[-A\sin x+B\cos x]+e^x[A\cos x+B\sin x]$
$\frac{dy}{dx}=e^x[-A\sin x+B\cos x]+y$ ...(1)
Again differentiating with respect to $x$, we get
$\frac{d^{2}y}{d{x}^2}=e^x[-A\sin x+B\cos x]+e^x[-A\cos x-B\sin x]+\frac{dy}{dx}$
$\frac{d^{2}y}{d{x}^2}=\left(\frac{dy}{dx}-y\right)y+\frac{dy}{dx}$ [using (1)]
$\frac{d^{2}y}{dx}-2\frac{dy}{dx}+2y=0$