Question

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) \(\frac{d^{2} y}{d x^{2}}-9 x=13\)
(B) \(\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0\)
(C) \(\frac{d^{2} y}{d x^{2}}+3 y=4\)
(D) \(\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}-x y=0\)

Answer 1

Ans: (B)
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:-
$$
\begin{aligned}
y^{\prime \prime} &=y^{\prime}+\left(y^{\prime}-y\right)+e^{x}(-A \cos x-B \sin x) \\
&=2 y^{\prime}-y-y \quad \Rightarrow y^{\prime \prime}-2 y^{\prime}+2 y=0
\end{aligned}
$$

Answer 2

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Answer 3

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