General solution of $\frac{d^2 y}{d x^2}

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Differential Equations

Question:

General solution of $\frac{d^2 y}{d x^2}=e^{-2 x}$ is :

Options:

$y=\frac{1}{4} e^{-2 x}+c$

$y=e^{-2 x}+c x+d$

$y=\frac{1}{4} e^{-2 x}+c x+d$

$y=e^{-2 x}+c x^2+d$

Correct Answer:

$y=\frac{1}{4} e^{-2 x}+c x+d$

Explanation:

$\frac{d^2 y}{d x^2}=e^{-2 x}, \frac{d y}{d x}=\frac{e^{-2 x}}{2}+k_1$

Integrating, $y=\frac{e^{-2 x}}{4}+k_1 x+k_2 \Rightarrow y=\frac{e^{-2 x}}{4}+cx+d$

Hence (3) is the correct answer.