The PDE is known to be integrable in steps if .
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For , Laplace returns NULL since the default number of iterations is 5.
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To obtain the solution in this example use the optional argument numberofiterations.
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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.
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Let's show that if is a solution to , then is a solution to .
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Also, if is a solution to , then is a solution to .
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Finally, if is a solution to , then is a solution to .
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Now, remarkably, we start with the zero solution to , integrate the equation to find and apply to find :
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So this is the solution to
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A similar sequence of PDEs and transformations can be constructed to find a solution depending on an arbitrary function of y.