Question

dy/dx=y³-2y²x-yx²/yx²+x³
Homogeneous differential equations

Answer 1

Answer:

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yx^2.dx=(x^3+y^3)dy

dy/dx=yx^2/(x^3+y^3) , let y=v.x

dy/dx=v.1+x.dv/dx

v+x.dv/dx=vx^3/(x^3+v^3.x^3)

v+x.dv/dx=v/(1+v^3)

x.dv/dx=v/(1+v^3) -v=(v-v-v^4)/(1+v^3)

x.dv/dx= -v^4/(1+v^3)

(1+v^3)/v^4.dv=-1/x.dx

[(1/v^4)+(1/v)].dv= -1/x.dx , integrate both side

-4.(1/v^5)+log v=-log x+C

log v+log x=C+4/v^5

logv.x=C+4/v^5

v.x=e^(C+4/v^5) , put v=y/x

y=e^(C+4.x^5/y^5) , Answer

Step-by-step explanation: