Solve (y+sqrt{x^2}+y^2)dx-xdy=0

$$( y + \sqrt { x ^ { 2 } } + y ^ { 2 } ) d x - x d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

$$ydx+\sqrt{x^{2}}dx+y^{2}dx-xdy=0$$

Combine ydx and -xdy to get 0.

$$\sqrt{x^{2}}dx+y^{2}dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{2}+x\sqrt{x^{2}}\right)d=0$$

Divide 0 by \sqrt{x^{2}}x+y^{2}x.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$