The general solution of the differential equation x 2 ( 1 + y 3 ) d x = y 2 ( 1 + x 3 ) d y is

On rearranging the given equation, we reduce the equation in the form. 

$\frac{x^{2}dx}{1+x^3}=\frac{y^{2}dy}{1+y^3}$

Integrating both side, using method of substitution 

let $(1+x^3)=t,$ on differentiating both sides we get $3x^2=\frac{dt}{dx}\Rightarrow x^{2}dx=\frac{dt}{3}$ 

Now,$\int \frac{x^{2}dx}{1+x^3}=\frac{1}{3}\int \frac{dt}{t}=\frac{1}{3}\ln t$

$$ Doing similarly for right-hand side we obtain,$\frac{1}{3}\ln (1+x^3)=\frac{1}{3}\ln (1+y^3)+\frac{\ln C}{3}$

$\ln \left(1+x^3\right)=\ln \left(c(1+y^3\right))$

$(1+x^3)=\left(c\right(1+y^3\left)\right)$