The teacher hasn't uploaded a solution for this question yet.
The given differential equation is: $x^{2}y~dx-(x^{3}+y^{3})dy=0$ Rewrite it as: $x^{2}y~dx = (x^{3}+y^{3})dy$ $\frac{dy}{dx} = \frac{x^{2}y}{x^{3}+y^{3}}$
The equation is homogeneous because each term has a degree of 3. Let $y = vx$, so $\frac{dy}{dx} = v + x\frac{dv}{dx}$.
Substitute $y = vx$ into the differential equation: $v + x\frac{dv}{dx} = \frac{x^{2}(vx)}{x^{3}+(vx)^{3}}$ $v + x\frac{dv}{dx} = \frac{vx^{3}}{x^{3}+v^{3}x^{3}}$ $v + x\frac{dv}{dx} = \frac{v}{1+v^{3}}$ $x\frac{dv}{dx} = \frac{v}{1+v^{3}} - v$ $x\frac{dv}{dx} = \frac{v - v(1+v^{3})}{1+v^{3}}$ $x\frac{dv}{dx} = \frac{v - v - v^{4}}{1+v^{3}}$ $x\frac{dv}{dx} = \frac{-v^{4}}{1+v^{3}}$
Separate the variables: $\frac{1+v^{3}}{v^{4}}dv = -\frac{dx}{x}$ Integrate both sides: $\int \frac{1+v^{3}}{v^{4}}dv = \int -\frac{1}{x}dx$ $\int (\frac{1}{v^{4}} + \frac{v^{3}}{v^{4}})dv = -\int \frac{1}{x}dx$ $\int (v^{-4} + v^{-1})dv = -\int \frac{1}{x}dx$ $\frac{v^{-3}}{-3} + \ln|v| = -\ln|x| + C$ $-\frac{1}{3v^{3}} + \ln|v| = -\ln|x| + C$
Substitute $v = \frac{y}{x}$: $-\frac{1}{3(\frac{y}{x})^{3}} + \ln|\frac{y}{x}| = -\ln|x| + C$ $-\frac{x^{3}}{3y^{3}} + \ln|y| - \ln|x| = -\ln|x| + C$ $-\frac{x^{3}}{3y^{3}} + \ln|y| = C$ $\ln|y| = \frac{x^{3}}{3y^{3}} + C$
The solution to the differential equation is: $\ln|y| = \frac{x^{3}}{3y^{3}} + C$
Final Answer: $\ln|y| = \frac{x^{3}}{3y^{3}} + C$
AI generated content. Review strictly for academic accuracy.