Solve e^y(1+x^2)dy=2x(1

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

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{2x}{y\left(x^{2}+1\right)e^{y}}\text{, }&x\neq -i\text{ and }x\neq i\text{ and }y\neq 0\\d\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(e^{y}+e^{y}x^{2}\right)dy=2x\times 1$$

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

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

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

$$e^{y}dy+e^{y}x^{2}dy=2x\times 1$$

Multiply 2 and 1 to get 2.

$$e^{y}dy+e^{y}x^{2}dy=2x$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide both sides by e^{y}x^{2}y+e^{y}y.

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

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

$$d=\frac{2x}{yx^{2}e^{y}+ye^{y}}$$

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

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

Solve for d

$$\left\{\begin{matrix}d=\frac{2x}{y\left(x^{2}+1\right)e^{y}}\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(e^{y}+e^{y}x^{2}\right)dy=2x\times 1$$

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

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

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

$$e^{y}dy+e^{y}x^{2}dy=2x\times 1$$

Multiply 2 and 1 to get 2.

$$e^{y}dy+e^{y}x^{2}dy=2x$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide both sides by e^{y}x^{2}y+e^{y}y.

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

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

$$d=\frac{2x}{yx^{2}e^{y}+ye^{y}}$$

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1-\left(dy\right)^{2}e^{2y}}+1}{dye^{y}}\text{; }x=\frac{-\sqrt{1-\left(dy\right)^{2}e^{2y}}+1}{dye^{y}}\text{, }&d\neq 0\text{ and }y\neq 0\\x=\frac{dye^{y}}{2}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{1-\left(dy\right)^{2}e^{2y}}+1}{dye^{y}}\text{; }x=\frac{-\sqrt{1-\left(dy\right)^{2}e^{2y}}+1}{dye^{y}}\text{, }&d\neq 0\text{ and }y\neq 0\text{ and }|d|\leq \frac{1}{|y|e^{y}}\\x=\frac{dye^{y}}{2}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$