Solve differential equation xydy−y^2dx= (x+y)^2 e^(−y/x)

Elleanor Mckenzie 2020-11-07 Answered
Solve differential equation xydyy2dx=(x+y)2e(y/x)
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Expert Answer

jlo2niT
Answered 2020-11-08 Author has 96 answers

xydyy2dx=(x+y)2e(y/x)
xydy/dxy2=(x+y)2e(y/x) (1)
Put y/x=t=>dy/dx=y+x(dt)/dx
Now the equation (1) becomes
xxt[t+x(dt)/dx]x2t2=x2(1+t)2et
t[t+x(dt)/dx]t2=(1+t)2et
xt(dt)/dx=(1+t)2et
(ettdt)/(1+t)2=dx/x
(et((t+1)1)dt)/(1+t)2=dx/x
et(1/(1+t)1/(1+t)2)dt=dx/x (2)
Integrating on the both sides
et(1/(1+t)1/(1+t)2)dt=dx/x
Use the standard formula of integration
ex(f(x)+f(x))dx=exf(x)+C
et/((1+t))=ln(x)+log(C)
ey/x/(1+y/x)=ln(Cx)
ey/x=(1+y/x)ln(Cx)

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Jeffrey Jordon
Answered 2021-12-25 Author has 2087 answers

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