<s>[INST] General solution of differntial equation $(x^2)(y)dx = ( (x^3) + (y^3))dy$ I tried this problem by many ways , i took $dy$ to other side and tried to form bernaulli equation but it is not helping and not giving me proper answer .
 [/INST] Notice, we have $$x^2ydx=(x^3+y^3)dy$$
$$\frac{dy}{dx}=\frac{x^2y}{x^3+y^3}$$
$$\frac{dy}{dx}=\frac{\frac{y}{x}}{1+\frac{y^3}{x^3}}$$
Now, let $y=vx\implies \frac{dy}{dx}=v+x\frac{dv}{dx}$
$$v+x\frac{dv}{dx}=\frac{v}{1+v^3}$$
$$x\frac{dv}{dx}=\frac{-v^4}{1+v^3}$$
$$\frac{1+v^3}{v^4}\ dv=-\frac{dx}{x}$$
$$\left(\frac{1}{v^{4}}+\frac{1}{v}\right)\ dv=-\frac{dx}{x}$$
$$\int\left(\frac{1}{v^{4}}+\frac{1}{v}\right)\ dv=-\int \frac{dx}{x}$$
$$\frac{-1}{3v^3}+\ln|v|=-\ln|x|+C$$
$$\frac{x^3}{3y^3}-\ln\left|\frac{y}{x}\right|=\ln|x|+C_1$$
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