Solve 2ydy=(x^2+1)dx

$$2 y d y = ( x ^ { 2 } + 1 ) d x$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=\sqrt[3]{\sqrt{y^{4}+\frac{1}{27}}+y^{2}}+\sqrt[3]{-\sqrt{y^{4}+\frac{1}{27}}+y^{2}}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$2y^{2}d=\left(x^{2}+1\right)dx$$

Use the distributive property to multiply x^{2}+1 by d.

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

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

$$2y^{2}d=dx^{3}+dx$$

Subtract dx^{3} from both sides.

$$2y^{2}d-dx^{3}=dx$$

Subtract dx from both sides.

$$2y^{2}d-dx^{3}-dx=0$$

Reorder the terms.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{\sqrt[3]{3\sqrt{81y^{4}+3}+27y^{2}}+\sqrt[3]{-3\sqrt{81y^{4}+3}+27y^{2}}}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$