How do you solve this differential equation: Sqrt (1+x^2) dy + sqrt (1+y^2) dx = 0?

Answer 1

Go ahead and isolate the x & dx expressions on one side and the y & dy expressions on the other side.

You get an integral that looks like dy/(1+y²) on one side & a negative integral on the other.

Then use the triangle method to change the variables to trig functions.

You will then get the integral of sec(theta). That integral is ln(sec(theta) + tan(theta))

Then substitute the x & y expressions back and get something like

ln( sqrt(1+y²) + y) = -ln( sqrt(1+x²) + x) + constant

You can simplify from there!