Question

Find the solution of the differential equation: $x\sqrt{1+y^2}dx+y\sqrt{1+x^2}dy=0.$

Answer 1

Given $x\sqrt{1+y^2}dx+y\sqrt{1+x^2}dy=0\Rightarrow \frac{1}{2}\int \frac{2xdx}{\sqrt{1+x^2}}+\frac{1}{2}\int \frac{2ydy}{\sqrt{1+y^2}}=0\Rightarrow \frac{1}{2}\times 2\sqrt{1+y^2}=C$ i.e., $\sqrt{1+x^2}+\sqrt{1+y^2}=C$.