Solve xdx=(x^2/y-y^3)dy

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

Solve for d

$$d=0$$

$$y\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

$$xdxy=\left(\frac{x^{2}}{y}-y^{3}\right)dyy$$

Multiply x and x to get x^{2}.

$$x^{2}dy=\left(\frac{x^{2}}{y}-y^{3}\right)dyy$$

Multiply y and y to get y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{3} times \frac{y}{y}.

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

Since \frac{x^{2}}{y} and \frac{y^{3}y}{y} have the same denominator, subtract them by subtracting their numerators.

$$x^{2}dy=\frac{x^{2}-y^{3}y}{y}dy^{2}$$

Do the multiplications in x^{2}-y^{3}y.

$$x^{2}dy=\frac{x^{2}-y^{4}}{y}dy^{2}$$

Express \frac{x^{2}-y^{4}}{y}d as a single fraction.

$$x^{2}dy=\frac{\left(x^{2}-y^{4}\right)d}{y}y^{2}$$

Express \frac{\left(x^{2}-y^{4}\right)d}{y}y^{2} as a single fraction.

$$x^{2}dy=\frac{\left(x^{2}-y^{4}\right)dy^{2}}{y}$$

Cancel out y in both numerator and denominator.

$$x^{2}dy=dy\left(x^{2}-y^{4}\right)$$

Use the distributive property to multiply dy by x^{2}-y^{4}.

$$x^{2}dy=dyx^{2}-dy^{5}$$

Subtract dyx^{2} from both sides.

$$x^{2}dy-dyx^{2}=-dy^{5}$$

Combine x^{2}dy and -dyx^{2} to get 0.

$$0=-dy^{5}$$

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

$$-dy^{5}=0$$

The equation is in standard form.

$$\left(-y^{5}\right)d=0$$

Divide 0 by -y^{5}.

$$d=0$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

$$d=0\text{ and }y\neq 0$$

Solve for x

$$x\in \mathrm{R}$$

$$d=0\text{ and }y\neq 0$$