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Which of the following is solution of diff eqn $\frac{d^{2}y}{dx^{2}}+y=0$
Option: 1
$y=ae^{x}+be^{-x}$

Option: 2
$y=asinx+bcosx$

Option: 3
$y=ae^{x}+b$

Option: 4
$y=ae^{x}+be^{2x}$

Answers (1)

best_answer

As we learnt

Solution of differential equations -

A function y =f(x) is a solution of differential equation, if the substitution of f(x) and its derivative (s) in differential equation reduces it to an identity.

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For (A) $\rightarrow y=ae^x+be^{-x}$

$\Rightarrow \frac{d^{2}y}{dx^{2}}=ae^{x}+be^{-x}$ but they dont satisfy the given diff eqn.

For (B): $y= a\sin x+b\cos x\rightarrow$ $\frac{dy}{dx}=a\cos x-b\sin x$

$\frac{d^{2}y}{dx^{}2}=-a\sin x-b\cos x$ which satisfies the given diff eqn.

(B) is solution of diffrential equation

$For\: (c)\rightarrow y =ae^{x}+b \Rightarrow \frac{dy}{dx}=ae^{x}$

$\Rightarrow \frac{d^{2}y}{dx^{2}}=ae^{x}$

$y\: \: \: l\: \: \frac{d^{2}y}{dx^{2}}$ don't satisfy diffrential equation

$For\: (D)\rightarrow y =ae^{x}+be^{2x} \Rightarrow \frac{dy}{dx}=ae^{x}+2be^{2x}$

$\Rightarrow \frac{d^{2}y}{dx^{2}}=ae^{x}+4be^{2x}$

$y\: \: \: l\: \: \frac{d^{2}y}{dx^{2}}$ don't satisfy diffrential equation

So only option (B) is satisfies.

Posted by

Suraj Bhandari

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