solving a Pfaffian equation for a complete integral

Hello. Thank you for taking the time to help me. I cannot use mathematical symbols, thus, * will denote a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. To simplify things, I will let p=u*x and q=u*y. Furthermore, I will use ^ to denote a power. For example, x^2 means x squared. Also, / means division. The following is my question: The PDE is: (1+q^2)u = xp 1. I need to find a complete integral to the PDE. I have found, using Charpit's method, that du = [(1+ae^2u)u^2]/x dx + ae^u dy. (Is this right?) This is where I need help! I cannot solve for u. In essence, I must solve du=P(x,y,u,a)dx+Q(x,y,u,a)dy to obtain the function g(x,y,u,a)=b whi... click for more

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... click for more

Cauchy problem

I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of us with respect to x. Furthermore, I will let u*x=p and u*y=q. Also, I will let ^ denote a power. For example, x^2 means x squared, and, I will let / denote division. Here is my problem: The PDE is: (1+q^2)u-xp=0. I need to find the solution which passes contains the curve x^2=2u and y=0. Now, I have found the initial curve to be (using the strip conditions): x=a(t)=t y=b(t)=0 u=c(t)=t^2/2 p=d(t)=t q=e(t)=the square root of [2-t] I have also found the characteristic system to be: dx/ds=-x dy/ds=2qu du/ds=-px+2(q^2)u dp/ds=p-p(1+q^2) ... click for more

Partial Differential Equation simplification

I am trying to simplify Y(y) using the boundary condition Y(1) = 0. Please find details on the attached file. Thanks

Dirichelet problem

Hello. Thank you for taking the time for looking at this problem. I am having some difficulty with this question, although I am completely familiar with the method of seperable variables, eigenvalue problems, and Laplace's equation for a circle. May you please help me with this question (please note that I am unable to use mathematical symbols. Thus, I have written everything out): I need to use separation of variables to solve Lapalace's equation in the annular sector: 1< r<2, 0< theta< pi/2, u(1,theta)= f(theta), u(2,theta)=0, u(r,0)=0, u(r,pi/2)=0 Thank you!