Q72 / 155 · WBJEE 2020 · Mathematics
The differential equation of the family of curves $y=e^{x}(A \cos x+B \sin x)$ where $A, B$ are arbitrary constants is
- A$\frac{d^{2} y}{d x^{2}}-9 x=13$
- Correct answerB$\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$
- C$\frac{d^{2} y}{d x^{2}}+3 y=4$
- D$\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}-x y=0$
Solution
Hint:
$y=e^{x}(A \cos x+B \sin x)$
Differentiating w.r.t. x:-
$y^{\prime}=y+e^{x}(-A \sin x+B \cos x)$
Differentiating w.r.t. $x$ once again:-
$\begin{array}{c}
y^{\prime \prime}=y^{\prime}+\left(y^{\prime}-y\right)+e^{x}(-A \cos x-B \sin x) \\
=2 y^{\prime}-y-y \quad \Rightarrow y^{\prime \prime}-2 y^{\prime}+2 y=0
\end{array}$
Hence, the correct answer is option: 'B'.