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

**L O S A L A M O S N A T I O N A L L A B O R A T O R Y***Operated by the University of California for the U.S. Department of Energy*