Charpit method

The Charpit method is a method for finding solutions of second-order partial differential equations in mathematics.

The second-order partial differential equation is $f(x,y,z,p,q)=0$ (1) $F(x,y,z,p,q)=0$. (2)

Then the general solution is $dz=pdx+qdy$. (3)

The integral surfaces of eqn.(1)&(2) are obtained by eliminating p and q . Substitution of p and q in eqn. (3) provides the solution, which is obtained by characteristics equations.

${dp}/{(\partial f/\partial x+p\partial f\partial z)}={dq}/{(\partial f/\partial y+q\partial f\partial z)}$ = ${dz}/{-(p\partial f/\partial p+q\partial f\partial q)}$ = ${-dx}/{(\partial f/\partial p)}={-dy}/{\partial f\partial q}={dF}/0$