Question

Solve the differential equation by separation of variables. dx/dy= dy/dx=x^2*cuberoot(y+1)

Answer 1

`(2)/(3)(1+y)^{(2)/(3)}+ (x^3)/(3)=c`

Explanation:

variable separable form:

`\frac{\mathrm{d} y}{\mathrm{d} x}=x^2\sqrt[3] {1+y}\\(dy)/(√(1+y)) = x^2dx\\\int\frac{dy}{\sqrt[3] {1+y}} = \int x^2dx\\(2)/(3)(1+y)^{(2)/(3)}= (x^3)/(3)+c\\(2)/(3)(1+y)^{(2)/(3)}+ (x^3)/(3)=c`

hence the solution comes out be

`(2)/(3)(1+y)^{(2)/(3)}+ (x^3)/(3)=c`