Solve differential equation y'+y\cot(x)=sin(2x)

ka1leE 2020-11-10 Answered
Solve differential equationy+ycot(x)=sin(2x)
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Expert Answer

Arham Warner
Answered 2020-11-11 Author has 102 answers
y+P(x)y=Q(x)
I.F.=ePdx
y(I.F.)=(I.F.)Q(x)dx+c
P(x)=cot(x), Q(x)dx+c
I.F.=ecot(x)dx=elog(sin(x))=sin(x)
y(sin(x))=sin(x)sin(2x)dx+c
y(sin(x))=sin(x)2sin(x)cos(x)dx+c
y(sin(x))=2sin2(x)cos(x)dx+c
Fisrt solve the integral
Let
sin(x)=tcos(x)dx=dt
sin2(x)cos(x)dx=t2dt=t33+c1=sin3(x)3+c1
y(sin(x))=2sin3(x)3+c1+c
y(sin(x))=23sin3(x)+C, [c+c1=C]
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