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  • Which of the following is the general solution of frac{d^{2} y}{d x^{2}}-2 frac{d y}{d x} +y=0 ? (a) y=(A x+B) e^{x}

Which of the following is the general solution of \frac{d^{2} y}{d x^{2}}-2\left(\frac{d y}{d x}\right)+y=0 ?
(a) y=(A x+B) e^{x}
(b) y=(A x+B) e^{-x}
(c) y=A e^{x}+B e^{-x}
(d) y=A \cos x+B \sin x

Answers (1)

Ans: -

The answer is the option (a) y=(A x+B) e^{x}

Explanation: -

\begin{aligned} \quad \frac{d y}{d x} &=(A x+B) e^{x}+A e^{x}=(A x+A+B) e^{x} \\ \Rightarrow \quad & \frac{d^{2} y}{d x^{2}}=(A x+A+B) e^{x}+A e^{x}=(A x+2 A+B) e^{x} \\ \therefore \quad & \frac{d^{2} y}{d x^{2}}-2\left(\frac{d y}{d x}\right)+y =(A x+2 A+B) e^{x}-2(A x+A+B) e^{x}+(A x+B) e^{x} =0 \end{aligned}

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