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Maths-

General

Easy

Question

If $y equals e to the power of negative x end exponent left parenthesis A cos space x plus B sin space x right parenthesis$ then y satisfies

$\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} = 0$
$\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 2 y = 0$
$\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} + 2 y = 0$
$\frac{d^{2} y}{d x^{2}} + 2 y = 0$

hint Hint:

In this question we have to differentiate the function twice to get the required result.

The correct answer is: $fraction numerator d squared y over denominator d x squared end fraction plus 2 fraction numerator d y over denominator d x end fraction plus 2 y equals 0$

$I f space y equals e to the power of negative x end exponent left parenthesis A cos space x plus B sin space x right parenthesis d i f f e r e n t i a t i n g space b o t h space s i d e s comma space w e space g e t fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction e to the power of negative x end exponent left parenthesis A cos space x plus B sin space x right parenthesis rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative e to the power of negative x end exponent open parentheses A cos x plus B sin x close parentheses plus e to the power of negative x end exponent open parentheses negative A sin x plus B cos x close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative y plus e to the power of negative x end exponent open parentheses negative A sin x plus B cos x close parentheses A g a i n comma space d i f f e r e n t i a t i n g space b o t h space s i d e s comma space w e space g e t rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator negative d y over denominator d x end fraction plus e to the power of negative x end exponent open parentheses negative A cos x minus B sin x close parentheses minus e to the power of negative x end exponent open parentheses negative A sin x plus B cos x close parentheses rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction plus fraction numerator d y over denominator d x end fraction equals negative e to the power of negative x end exponent open parentheses A cos x plus B sin x close parentheses minus e to the power of negative x end exponent open parentheses negative A sin x plus B cos x close parentheses rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction plus fraction numerator d y over denominator d x end fraction equals negative y minus y minus fraction numerator d y over denominator d x end fraction rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction plus 2 fraction numerator d y over denominator d x end fraction plus 2 y equals 0$

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