Solve sqrt{1+x^2}dy+sqrt{1+y^2}dx=0

$$\sqrt { 1 + x ^ { 2 } } d y + \sqrt { 1 + y ^ { 2 } } d x = 0$$

Solve for d

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

Steps for Solving Linear Equation

Combine all terms containing d.

$$\left(\sqrt{1+x^{2}}y+\sqrt{1+y^{2}}x\right)d=0$$

The equation is in standard form.

$$\left(x\sqrt{y^{2}+1}+y\sqrt{x^{2}+1}\right)d=0$$

Divide 0 by \sqrt{1+x^{2}}y+\sqrt{1+y^{2}}x.

$$d=0$$

Solve for x

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