Solve xdx-(5y^4+3)dy=0

$$x d x - ( 5 y ^ { 4 } + 3 ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-\sqrt{5y^{5}+3y}\text{ or }x=\sqrt{5y^{5}+3y}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply 5y^{4}+3 by d.

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

Use the distributive property to multiply 5y^{4}d+3d by y.

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

To find the opposite of 5dy^{5}+3dy, find the opposite of each term.

$$x^{2}d-5dy^{5}-3dy=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y\geq 0\text{ and }|x|=\sqrt{y\left(5y^{4}+3\right)}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply 5y^{4}+3 by d.

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

Use the distributive property to multiply 5y^{4}d+3d by y.

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

To find the opposite of 5dy^{5}+3dy, find the opposite of each term.

$$x^{2}d-5dy^{5}-3dy=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\sqrt{y\left(5y^{4}+3\right)}\text{; }x=\sqrt{y\left(5y^{4}+3\right)}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{y\left(5y^{4}+3\right)}\text{; }x=-\sqrt{y\left(5y^{4}+3\right)}\text{, }&y\geq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$