Solve frac{d(x^{2}-y^{2})}{x^{2}-y^{3}}=d(x^2-y^2)/x^3-4^3

$$\frac { d ( x ^ { 2 } - y ^ { 2 } ) } { x ^ { 2 } - y ^ { 3 } } = \frac { d ( x ^ { 2 } - y ^ { 2 } ) } { x ^ { 3 } - 4 ^ { 3 } }$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq x^{\frac{2}{3}}\text{ and }x\neq 4\\d\in \mathrm{R}\text{, }&\left(y=\sqrt[3]{64+x^{2}-x^{3}}\text{ and }x\neq 4\right)\text{ or }\left(x=y\text{ and }y\neq 1\text{ and }y\neq 0\text{ and }y\neq 4\right)\text{ or }\left(x=-y\text{ and }y\neq 1\text{ and }y\neq 0\text{ and }y\neq -4\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x^{2}+4x+16\right)\left(-x^{2}+y^{3}\right), the least common multiple of x^{2}-y^{3},x^{3}-4^{3}.

$$\left(64-x^{3}\right)d\left(x^{2}-y^{2}\right)=\left(-x^{2}+y^{3}\right)d\left(x^{2}-y^{2}\right)$$

Use the distributive property to multiply 64-x^{3} by d.

$$\left(64d-x^{3}d\right)\left(x^{2}-y^{2}\right)=\left(-x^{2}+y^{3}\right)d\left(x^{2}-y^{2}\right)$$

Use the distributive property to multiply 64d-x^{3}d by x^{2}-y^{2}.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}=\left(-x^{2}+y^{3}\right)d\left(x^{2}-y^{2}\right)$$

Use the distributive property to multiply -x^{2}+y^{3} by d.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}=\left(-x^{2}d+y^{3}d\right)\left(x^{2}-y^{2}\right)$$

Use the distributive property to multiply -x^{2}d+y^{3}d by x^{2}-y^{2}.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}=-dx^{4}+x^{2}dy^{2}+y^{3}dx^{2}-dy^{5}$$

Add dx^{4} to both sides.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}+dx^{4}=x^{2}dy^{2}+y^{3}dx^{2}-dy^{5}$$

Subtract x^{2}dy^{2} from both sides.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}+dx^{4}-x^{2}dy^{2}=y^{3}dx^{2}-dy^{5}$$

Subtract y^{3}dx^{2} from both sides.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}+dx^{4}-x^{2}dy^{2}-y^{3}dx^{2}=-dy^{5}$$

Add dy^{5} to both sides.

$$64dx^{2}-64dy^{2}-dx^{5}+x^{3}dy^{2}+dx^{4}-x^{2}dy^{2}-y^{3}dx^{2}+dy^{5}=0$$

Reorder the terms.

$$-dx^{5}+dx^{4}-dx^{2}y^{3}-dx^{2}y^{2}+64dx^{2}+dy^{5}+dy^{2}x^{3}-64dy^{2}=0$$

Combine all terms containing d.

$$\left(-x^{5}+x^{4}-x^{2}y^{3}-x^{2}y^{2}+64x^{2}+y^{5}+y^{2}x^{3}-64y^{2}\right)d=0$$

Divide 0 by -x^{5}+x^{4}-x^{2}y^{3}-x^{2}y^{2}+64x^{2}+y^{5}+y^{2}x^{3}-64y^{2}.

$$d=0$$