Solve x(x+y)(y)-x^2=K(x^3+y^3)

$$x ( x + y ) ( y ) - x ^ { 2 } = K ( x ^ { 3 } + y ^ { 3 } )$$

Solve for K

$$\left\{\begin{matrix}K=\frac{x\left(xy-x+y^{2}\right)}{x^{3}+y^{3}}\text{, }&x\neq -y\\K\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by x+y.

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

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

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

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

$$x^{2}y+xy^{2}-x^{2}=Kx^{3}+Ky^{3}$$

Swap sides so that all variable terms are on the left hand side.

$$Kx^{3}+Ky^{3}=x^{2}y+xy^{2}-x^{2}$$

Combine all terms containing K.

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

The equation is in standard form.

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

Divide both sides by x^{3}+y^{3}.

$$\frac{\left(x^{3}+y^{3}\right)K}{x^{3}+y^{3}}=\frac{x\left(xy-x+y^{2}\right)}{x^{3}+y^{3}}$$

Dividing by x^{3}+y^{3} undoes the multiplication by x^{3}+y^{3}.

$$K=\frac{x\left(xy-x+y^{2}\right)}{x^{3}+y^{3}}$$

Divide x\left(-x+y^{2}+yx\right) by x^{3}+y^{3}.

$$K=\frac{x\left(xy-x+y^{2}\right)}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$$