Solve ydx+(x^3y^3+x^3)dy=0

$$y d x + ( x ^ { 3 } y ^ { 3 } + x ^ { 3 } ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }y=\frac{\sqrt[3]{-x^{2}-1}}{x^{\frac{2}{3}}}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{3}y^{3}+x^{3} by d.

$$ydx+\left(x^{3}y^{3}d+x^{3}d\right)y=0$$

Use the distributive property to multiply x^{3}y^{3}d+x^{3}d by y.

$$ydx+x^{3}dy^{4}+x^{3}dy=0$$

Combine all terms containing d.

$$\left(yx+x^{3}y^{4}+x^{3}y\right)d=0$$

The equation is in standard form.

$$\left(x^{3}y^{4}+xy+yx^{3}\right)d=0$$

Divide 0 by yx+x^{3}y^{4}+x^{3}y.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\sqrt{-\frac{1}{y^{3}+1}}\text{; }x=-\sqrt{-\frac{1}{y^{3}+1}}\text{, }&y<-1\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$