How to separate variable equation #dy/dx = (xy+x)^(2/3) #?

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Answer 1 · 2018-01-21T15:42:07

#y=(1/5x^(5/3)+C/3)^3-1#

We have the differential equation:

#dy/dx=(xy+x)^(2/3)#

Rewrite the equation to the form #dy/dx=h(x)*g(y)#

#dy/dx=(x*(y+1))^(2/3)#

#dy/dx=x^(2/3)(y+1)^(2/3)#

Separate the variables

#(y+1)^(-2/3)dy=x^(2/3)dx#

Integrate both sides

#int(y+1)^(-2/3)dy=intx^(2/3)dx#

#3(y+1)^(1/3)=3/5x^(5/3)+C#

Solve for y

#y=(1/5x^(5/3)+C/3)^3-1#