Question

The differential equation whose solution is y= acos(x+2)

Answer 1

Question :-

Find the differential equation whose solution is y= acos(x+2) .

Answer :-

$\rightarrow \: \red{ \boxed{ \sf{ \frac{ {d}^{2} y}{d \: {x}^{2} } \: + y = 0}} }\: \:$

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To Find :-

Find differential equation.

Step - by - step explanation :-

Given solution is ,

$\implies \: \sf{ y \: = a \cos \big((x + 2) \big) } \\ \\ \sf{differentiate \: on \: both \: sides \: \: with \: } \\ \sf{respect \: to \: x \: } \\ \\ \rightarrow \sf{ \frac{dy}{dx} = - a \sin \big((x + 2) \big) \frac{d}{dx} (x + 2)} \\ \\ \rightarrow \: \sf{ \frac{dy}{dx} = - a \: \sin \big((x + 2) \big) } \\ \\ \sf{ to \: remove \: \red{a} \: again \: differentiate} \\ \sf{ with \: respect \: to \: x} \\ \\ \rightarrow \sf{ \frac{ {d}^{2} y}{d \: {x}^{2} } = - a \cos \big((x + 2) \big) \frac{d}{dx} (x + 2)} \\ \\ \because \: \bf{ y \: = a \: \cos \big((x + 2) \big) }\\ \\ \therefore \\ \rightarrow \: \sf{ \frac{ {d}^{2} y}{d \: {x}^{2} } \: = - (y)} \\ \\ \rightarrow \: \boxed{ \sf{ \frac{ {d}^{2} y}{d \: {x}^{2} } \: + y = 0}}$

This is the required differential equation.

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