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

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

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

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

Combine all terms containing d.

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

Divide both sides by 2x^{3}+x.

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

Dividing by 2x^{3}+x undoes the multiplication by 2x^{3}+x.

$$d=\frac{y}{2x^{3}+x}$$

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

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

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt[3]{\frac{6}{d|d|}}\left(\sqrt[3]{d\sqrt{3\left(27y^{2}+2d^{2}\right)}+9y|d|}+\sqrt[3]{-d\sqrt{3\left(27y^{2}+2d^{2}\right)}+9y|d|}\right)}{6}\text{, }&d\neq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }d=0\end{matrix}\right.$$