The PDE is known to be integrable in steps if .
For , Laplace returns NULL since the default number of iterations is 5.
To obtain the solution in this example use the optional argument numberofiterations.
We analyze here the case to show some of the details of the method. We define a sequence of three PDEs, , and . We wish to solve . The PDEs and are generated by the method of Laplace. We also define three maps which we denote by , and . These are also prescribed by the method of Laplace.
Let's show that if is a solution to , then is a solution to .
Also, if is a solution to , then is a solution to .
Finally, if is a solution to , then is a solution to .
Now, remarkably, we start with the zero solution to , integrate the equation to find and apply to find :
So this is the solution to
A similar sequence of PDEs and transformations can be constructed to find a solution depending on an arbitrary function of y.