Solve differential equation dy/dx+ycos(x)= 4cos(x), y(0)=6

Kye 2021-02-23 Answered

Solve differential equation dydx+ycos(x)=4cos(x), y(0)=6

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Expert Answer

Alannej
Answered 2021-02-24 Author has 104 answers

dydx+ycosx=4cosx
That is dydx+(cosx)y=4cosx
dydx+P(x)=Q(x)
So P(x)=cosx, Q(x)=4cosx
Integrating factor is
I.F.=e(P(x)dx)
=ecosxdx
=esinx
yI.F.=Q(x)I.F.dx+c
yesinx=4cosxesinxdx+c
=4esinxcosxdx+c
=4etdt+c (by subtitution)
=4et+c
yesinx=4esinx+c
y(x)=4esinx+c
Now apply the initial condition y(0)=6 in the general solution
4esin(0)+c=6
4e0+c=6=>4+c=6
c=6-4=2 Now substitute 2 for c in general solution
Thus,the particular solution is y(x)=4esinx+2

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