Solve differential equation y'+ycosx= sinxcox

Solve differential equation y'+ycosx= sinxcox

Question
Solve differential equation \(y'+ycosx= sinxcox\)

Answers (1)

2021-02-04
\(y'+P(x)y= Q(x)\)
\(I.F.= e^(int p(x)dx)\)
\(y I.F= int(Q(x)I.F.)dx\)
\(y'+ycosx= sinxcosx\)
\(P(x)= cosx\)
\(Q(x)= sinxcosx\)
\(I.F.= e^(int cosxdx)= e^(sinx)\)
\(y I.F.= int(Q(x)I.F.)dx\)
\(ye^(sinx)= int(sinxcosx e^(sinx))dx\)
Let \(sinx=u\)
\(=> cosxdx=du\)
\(ye^(sinx)= int ue^u du\)
\(= ue^u-e^u+C\)
\(= sin xe^(sinx)-e^(sinx)+C\)
\(=> ye^(sinx)= sin xe^(sinx)-e^(sinx)+C\)
\(=> y= sinx-1+Ce^(-sinx)\)
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