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  • Solve the differential equation dy = \cos x(2 - y\; cosec \;x) dx given that y = 2 when x=\frac{\pi}{2}

Solve the differential equation dy = \cos x(2 - y\; cosec \;x) dx given that y = 2 when x=\frac{\pi}{2}

Answers (1)

Given:
dy=\cos x(2-y\; cosec \;x)dx

\left ( \frac{\pi}{2},2 \right ) is a solution of the given differential equation

Rewriting the given equation

\frac{dy}{dx}=2\cos x-y \cot x\\ \frac{dy}{dx}+y \cot x=2\cos x\\

It is a first order differential equation

p(x)=\cot x\\ q(x)=2 \cos x

Calculate integrating factor

IF=e^{\int p(x)dx}\\ IF=e^{\int \cot x dx}\\ IF=e^{\ln \sin x}\\ Formula: \int \cot x =\ln \sin x\\ IF=\sin x

Therefore, the solution of the differential equation is

y.(IF)=\int q(x).(IF)dx\\ y \sin x= 2\int \cos x \sin x dx\\ y \sin x =\int \sin 2x \;dx\\ y \sin x = -\frac{1}{2}\cos 2x +c

Substituting \left ( \frac{\pi}{2},2 \right )to find the value of c

2= \frac{1}{2}+c\\ c=\frac{3}{2}

Hence the solution is

y \sin x=-\frac{1}{2}\cos 2x+\frac{3}{2}

Posted by

infoexpert24

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