# Find the solution of the Clairaut equation

How do you find the solution of the Clairaut equation $$u = xu_{x} + yu_{y} + \sqrt{1 + u^2_{x} + u^ 2_{y}}$$

with initial data

$$x(0, s) = a\cos(s)$$$$y(0, s) =a\cos(s)$$$$u(0, s) = 0,$$

where α is a constant such that α > 1.

• It seems the initial data is not consistent with your pde. What's the relation between $x$ and $y$ with $s$. Are we supposed to use the the method of characteristics? – minmax Nov 17 '18 at 22:03
• Have you tried to differentiate it with wrt $x$ and then wrt $y$? – minmax Nov 17 '18 at 22:07