DiffEqs`DETools`

This package provides the following functions for solving differential equations.

FOLDSolve[eqn, y[t], t]
    solves a first-order linear differential equation for y[t].
    
BernoulliDSolve[eqn, y[t], t]
    solves a Bernoulli-type differential equation for y[t].
    
RiccatiDSolve[eqn, y[t], t, z[t]]
    solves a Riccati-type differential equation for y[t], given a known solution z[t].
    
SeparableDSolve[]
    returns an implicit solution of a separable differential equation.

SeriesDSolve[eqn, y[t], t, t0 , n]
    returns the first n terms of a series expansion about t0 of the solution of a first- or second-order differential equation.


Finding solutions of differential equations.

 

This loads the package.

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FOLDSolve

A few examples of the use of FOLDSolve:

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Here’s an example on which DSolve works for a very long time.

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BernoulliDSolve

A few examples of the use of BernoulliDSolve:

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RiccatiDSolve

A few examples of the use of RiccatiDSolve:

The fourth argument is a known solution.

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Here’s an example on which DSolve fails.

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SeparableDSolve

A few examples of the use of SeparableDSolve:

The equation is given in separated form.

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The next example uses the InitialPoint option. Note that the first number is a value for the variable on the left side of the equation and the second number is a value for the variable on the right side of the equation. Here the initial condition is that y = 0 when t = 1.

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Since SeparableDSolve returns an equation, it can be combined easily with Solve.

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A result from SeparableDSolve is easy to feed into PlotImplicit from the DEGraphics package.

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SeriesDSolve

A few examples of the use of SeriesDSolve:

A nonlinear first-order initial-value problem:

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A nonlinear second-order initial-value problem:

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For a linear second-order equation (with no initial conditions), SeriesDSolve returns partial sums of a pair of “fundamental” solutions, if possible.

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Here we get expansions about a point other than 0.

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Nonpolynomial coefficients are no problem.

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If there is a nonhomogeneous term,  SeriesDSolve returns the “rest” solution (i.e., the solution satisfying zero initial conditions).

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Complex powers and Frobenius-series solutions are found as well.

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Converted by Mathematica      June 10, 2002