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

Answer 1

Answer: (A)

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