Differential Equations

We looked at the solve command when we discussed Algebraic Calculations. Maple can also solve differential equations with the dsolve command.

First off, define a differential equation in a similar way as you have been doing:

> sample_DE := x^2 * diff(y(x), x) + y(x)= exp(x);

                                2 /  d      \
                  sample_DE := x  |---- y(x)| + y(x) = exp(x)
                                  \ dx      /

Now we can solve the differential equation with dsolve:

> dsolve( sample_DE, y(x) );

                                   (x - 1) (x+ 1)
                             / exp(---------------)
                            |             x
           y(x) = exp(1/x)  | -------------------- dx + exp(1/x) _C1
                            |            2
                           /            x

Since we did not define initial conditions, Maple assigned a constant ( _C1 ) to the equation.
Here is another example, sample2_DE:

> sample2_DE := diff(y(u),u) + y(u)^2 +(2*u+1)*y(u) + u^2 + u + 1 =0;

                    /  d      \       2                    2
      sample2_DE := |---- y(u)| + y(u)  + (2 u+ 1) y(u) + u  + u + 1 = 0
                    \ du      /

We are going to define the initial conditions, initial, so that y(1)=1:

> initial := y(1) = 1;

                              initial := y(1)= 1

Now use dsolve to solve the differential equation given the initial conditions. Notice that the two definitions are in curly brackets, {}:

>  dsolve( {sample2_DE, initial}, y(u) );

                                         exp(- u)
                      y(u) = - u +----------------------
                                   3/2 exp(-1) - exp(- u)

We can simplify the above expression:

> simplify(");

                        - 3 u exp(-1) + 2 uexp(- u) + 2 exp(- u)
               y(u) = ------------------------------------------
                                 - 3 exp(-1) +2 exp(- u)

Solutions for equations can be calculated numerically or as a series of equations. Maple also has the capability of solving multiple order differential equations. For more information, look at one of the references listed at the end of this tutorial.


This page Maintained by Dale H. Leschnitzer
Last Modified Monday, November 4, 1996

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