Chaotic Junkyard

The one-dimensional wave equation is In such the characteristic equation By solving,  By partial differential, Similarly, Substitute back to the PDE, it results in: After arriving the general solution, consider the initial values By integrating the second equation, Therefore By substituting back to the general solution, 

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)