Solve (x^2-y^2)dx+(x^2+y^2)dy=0

$$( x ^ { 2 } - y ^ { 2 } ) d x + ( x ^ { 2 } + y ^ { 2 } ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=\frac{\sqrt[3]{3\sqrt{33}\left(|y|\right)^{3}-19y^{3}}+\sqrt[3]{-3\sqrt{33}\left(|y|\right)^{3}-19y^{3}}-y}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}-y^{2} by d.

$$\left(x^{2}d-y^{2}d\right)x+\left(x^{2}+y^{2}\right)dy=0$$

Use the distributive property to multiply x^{2}d-y^{2}d by x.

$$dx^{3}-y^{2}dx+\left(x^{2}+y^{2}\right)dy=0$$

Use the distributive property to multiply x^{2}+y^{2} by d.

$$dx^{3}-y^{2}dx+\left(x^{2}d+y^{2}d\right)y=0$$

Use the distributive property to multiply x^{2}d+y^{2}d by y.

$$dx^{3}-y^{2}dx+x^{2}dy+dy^{3}=0$$

Combine all terms containing d.

$$\left(x^{3}-y^{2}x+x^{2}y+y^{3}\right)d=0$$

The equation is in standard form.

$$\left(x^{3}-xy^{2}+y^{3}+yx^{2}\right)d=0$$

Divide 0 by x^{3}-y^{2}x+x^{2}y+y^{3}.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{\sqrt[3]{3\sqrt{33}\left(|y|\right)^{3}-19y^{3}}+\sqrt[3]{-3\sqrt{33}\left(|y|\right)^{3}-19y^{3}}-y}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$