Non linear PDE.
I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of u with respect to x. Moreover, I will further simplify things by letting p=u*x and q=u*y. Also, ^ denotes a power (for example, x^2 means x squared) and / denotes division. This is the problem:
The PDE is: xp+yq+p+q-pq=u
I need to find a complete integral. I should use Charpit's method, where I find p=P(x,y,u,a) and q=Q(x,y,u,a). I then solve du=P(x,y,u,a)dx+Q(x,y,u,a)dy to obtain f(x,y,u,a)=b, thus giving me u=u(x,y,a,b), a complete integral. Now I have already found the characteristic system to be:
dx/x+1-q = dy/y+1-p = du/u-pq = dp/0 = dq/0. (Is this right?) Please help me solve this system and find the appropriate P and Q for the Pfaffian equation du = Pdx+Qdy. Can you also help me solve this! Please help!!!
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- Partial differentiation.doc
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