Is the Problem Well-Posed?

As previously discussed, DSolve returns a general solution for a problem if no initial or boundary conditions are specified.

The general solution to this equation is returned.

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However, if initial or boundary conditions are specified, the output from DSolve must satisfy both the underlying differential equation as well as the given conditions.

Here is an example with a boundary condition.

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In such cases, it is useful to check whether DSolve has been asked a reasonable question--in other words, to check whether the problem is well-posed. An initial or boundary value problem is said to be well-posed if a solution for it is guaranteed to exist in some well-known class of functions (for example, analytic functions), if the solution is unique, and if the solution depends continuously on the data. Given an ODE of order n (or a system of n first-order equations) and n initial conditions, there are standard existence and uniqueness theorems that show that the problem is well-posed under a specified set of conditions. The right-hand side of the first-order linear ODE in the previous example is a polynomial in y[x] and hence infinitely differentiable. This is sufficient to apply Picard's existence and uniqueness theorem, which only requires that the right-hand side be Lipschitz-continuous.

Most problems that arise in practice are well-posed since they are derived from sound theoretical principles. However, as a note of caution, we would like to give a few examples where DSolve might have difficulty finding a satisfactory solution to the problem.

Here is the solution to a first-order ODE in which the right-hand side fails to satisfy the Lipschitz condition around 0.

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The general solution has two branches.

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DSolve succeeds in picking out the correct branch for the given initial condition.

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Here is a second-order ODE. The boundary conditions do not allow any solution to this problem.

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In this example, DSolve returns a pair of solutions. As the table shows, the first solution is only valid for values of x greater than or equal to 2.

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Finally, it is possible that a problem could have a solution but that DSolve could fail to find it because the general solution is in implicit form or involves higher transcendental functions.

In this example, a solution is available only after inverting the roles of the dependent and independent variables.

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This concludes the discussion of the basic principles for effectively working with DSolve. The next section contains a list of references that were found to be useful either during the development of DSolve or during the preparation of this documentation.