Solving a Pfaffian equation for a complete integral
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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 which gives u=u(x,y,a,b) a complete integral! I In this case, P(x,y,u,a) = [(1+ae^2u)u^2]/x and Q(x,y,u,a) = ae^u. Also, I asked another TA for help but she simply integrated [(1+ae^2u)u^2]/x with respect to x and ae^u with respect to y. This was incorrect since the equation in question is a Pfaffian equation or a differential. Thank you for your help!
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The expert solves a pfaffian equation for a complete integral.
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