Canonical Forms and General Solutions of 2nd PDE

Consider the second-order equation in which the derivatives of second-order all occur linearly, with coefficients only depending on the independent variables:

Step 1: Solving the characteristic equation is:

Step 2: Solve the 1^st order differential equation.

Step 3: Define μ(x, y) and η(x, y) as the two “constant” function as the result of the Step 2

Step 4: u = u(μ(x, y), η(x, y))

Step 5: Conduct partial differentiation and substitute back the result into the original PDE, resulting in a Canonical Form.

Step 6: Integrate (partial) the simple transformed Canonical Form to arrive General Solution

(Note that if all coefficients are Constants, final answer: u = μ(x, y) +  η(x, y), as the general solution)