Solve the differential equation x dy/dx= y+xe^(y/x), y=vx

Jason Farmer 2021-02-24 Answered
Solve the differential equation xdy/dx=y+xe(y/x), y=vx
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Expert Answer

krolaniaN
Answered 2021-02-25 Author has 86 answers

Divide both sides by x and further simplify it
x/xdy/dx=y/x+(xey/x)/x
dy/dx=y/x+e(y/x)
Substitute y=xv=>y/x=v
Differentiate with respect to x
dy/dx=v+x(dv)/dx
Hence, v+x(dv)/dx=v+ev
Substract v from both sides and further simplify it
v+x(dv)/dxv=v+evv
x(dv)/dx=ev
(dv)/ev=dx/x
evdv=dx/x
Integrate both sides
evdv=dx/x+c
ev=logx+logc
ev=log(xc)
ev=log(xc)
Taking log both sides
v=log(log(xc))
v=log(log(xc)) Substitute v=y/x
y/x=log(log(xc))
y=xlog(log(xc))

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