Question

Solve differential equationy'+ycosx= sinxcox

First order differential equations
ANSWERED
asked 2021-02-03

Solve differential equation \(y'+y\cos x= \sin x\cos x\)

Expert Answers (1)

2021-02-04

\(y'+P(x)y= Q(x)\)
\(I.F.= e^{\int p(x)dx}\)
\(y I.F= \int(Q(x)I.F.)dx\)
\(y'+y\cos x= \sin x\cos x\)
\(P(x)= \cos x\)
\(Q(x)= \sin x\cos x\)
\(I.F.= e^{\int cosxdx}= e^{\sin x}\)
\(y I.F.= \int(Q(x)I.F.)dx\)
\(ye^{\sin x}= \int(\sin x\cos x e^{\sin x})dx\)
Let \(\sin x=u\)
\(\Rightarrow \cos xdx=du\)
\(ye^{\sin x}= \int ue^u du\)
\(= ue^u-e^u+C\)
\(= \sin xe^{\sin x}-e^{\sin x}+C\)
\(\Rightarrow ye^{\sin x}= \sin xe^{\sin x}-e^{\sin x}+C\)
\(\Rightarrow y= \sin x-1+Ce^{-\sin x}\)

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