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.
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