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| (1) |
A general solution for ode[1] is not known. Therefore, in principle, it would be not possible to solve an initial value problem that contains it, for example, by using the following initial conditions.
However, suppose that you find a solution for ode[1] that has been computed by another means, for example
You can pass the initial conditions and the solution directly to the DEtools[IVPsol] command.
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For the following example, the general solution of the ODE involves the inverse of an elliptic integral, represented by the elliptic function JacobiSN.
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| (5) |
| (6) |
Consider the following initial conditions for ode[2]:
When solving the complete ODE-IVP, instead of directly using the general solution gensol[2], dsolve first searches for symmetries from where a particular solution could be computed. The computed solution could be easier to specialize to satisfy the given conditions y(0) = 1, D(y)(0) = 1, resulting in the following ODE-IVP solution.
If you use the following command to view the results of trying to specialize the general solution gensol[2], instead of using a short-hand form for the process using symmetries, the following results are returned. This is the solution returned in Maple 12.
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| (9) |
Yet another general solution, involving elliptic integrals themselves, is obtained using Lie symmetry methods to compute it; this is the solution returned in Maple 11 and earlier. It can also be specialized.
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| (10) |
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| (11) |
As illustrated, the solution returned by dsolve without extra arguments (that is, dsolve([ode[2], ic[2]])) is the simplest one. This solution is obtained by first computing a particular solution from the symmetries (see DEtools,particularsol).
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| (12) |
This particular solution, which depends on only one arbitrary constant , can also be specialized.
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