How to Solve a First-Order Nonlinear PDE using the Method of Characteristics?

[Eren10]

i have to solve this equation :

du/dx * du/dy = x*y

u(x,y) = x for y =0

with putting this equation in the form : F(x,y,u,du/dx,du/dy) = 0 . it can be solved.
But mine book does not explain how to do this, there are no examples.

Can someone help me ? or any links of examples on the internet?

 

[kosovtsov]

Your PDE already is in a such form!?

If you suppose that

u(x,y)=A(x)*B(y)

then substitution such u into PDE gives you after separation of variables

diff(A(x),x)*A(x)/x=1/(B(y)*diff(B(y),y)/y)=c ,

where c is a constant. Solutions of these two ODEs lead to particular solution

u(x,y) = (c*x^2+C2)^(1/2)*1/c*(c*(y^2+C1*c))^(1/2)

(among other things this solution allows to obtain the general sulution to your PDE!).
If now substitute y=0 into above solution, we find that it must be

C2=0, C1=1/c

so the required solution is as follows

u(x,y) = x*(y^2+1)^(1/2)

 

[Eren10]

Thank you for giving the answer. Mine mistake I had to tell, to solve this problem we had to use method of characteristics. I did it and found the same answer. so it is correct.