How to | Use Derivatives for Setting Up Differential Equations
Mathematica's functions for solving differential equations can be applied to many different classes of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), differential-algebraic equations (DAEs), and boundary value problems (BVPs). Using derivatives to set up these equations for solving in
Mathematica is essential.
For an ODE, you can express the derivative of a function
of
with respect to
in several ways.
The most common way is to use
D:
| Out[8]= |  |
You can specify the order of differentiation, which in this case is 1:
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Using
also gives the derivative
:
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Using
is shorthand for
Derivative:
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FullForm shows that
and
Derivative are equivalent:
Out[2]//FullForm= |
| |  |
Use
D to set up the ODE
for solving and then store the equation as
:
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Use
TraditionalForm to see the ODE as it would appear in a mathematics textbook or journal article:
Out[7]//TraditionalForm= |
| |  |
Once the ODE is set up, use
DSolve to solve it symbolically:
| Out[8]= |  |
The solution
is represented as a rule in a nested list. For information on getting this solution out of the list and using it, see
How to: Use Rule Solutions.
Most of the time, ODEs are accompanied by boundary and initial conditions. Thus, evaluation of derivatives of functions for specific values of variables needs to be used frequently. This can be done in several ways. Here,
is used as an example.
Use
(shorthand for
ReplaceAll) to substitute
for
.
Mathematica first evaluates
D, then performs the replacement:
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Or, just evaluate
directly by using
:
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Also equivalent is
Derivative:
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As another example, set up the ODE
,
,
, for solving:
| Out[1]= |  |
Use
DSolve to obtain the symbolic solution:
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Of course, you can solve directly without setting up:
| Out[4]= |  |
Alternatively, you can express the ODE as a PDE and solve it for
instead of
:
| Out[17]= |  |
| Out[19]= |  |
Here, the ODE
,
is set up:
| Out[4]= |  |
Use
DSolve to obtain the symbolic solution:
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Instead of using
DSolve, you can use
NDSolve to obtain a numerical solution that corresponds to specific values of
that are involved in one of the coefficients of the ODE.
The following example gives the solution when
.
NDSolve gives results in terms of
InterpolatingFunction objects:
| Out[6]= |  |
The following function
gives numerical solutions for each value of
:
Plot the results at
and
.
Evaluate is used so that
evaluates properly in
Plot:
| Out[34]= |  |
Mathematica can also set up and solve PDEs.
The function
is used here to show the different ways partial derivatives can be expressed in
Mathematica.
The most common way to get a partial derivative is to use
D:
| Out[6]= |  |
| Out[7]= |  |
| Out[8]= |  |
You can accomplish the same thing by using slightly different syntax for
D:
| Out[11]= |  |
| Out[12]= |  |
| Out[13]= |  |
Alternatively, use
Derivative:
| Out[12]= |  |
| Out[13]= |  |
| Out[14]= |  |
FullForm shows that these methods are equivalent:
Out[15]//FullForm= |
| |  |
Next, consider several variations of a PDE.
Set up the PDE
for solving:
| Out[17]= |  |
Use
DSolve to find a symbolic solution to the PDE. The result is given in terms of an arbitrary function
C
:
| Out[19]= |  |
The solution can also be obtained in terms of a pure function.
Function represents a pure function in
Mathematica:
| Out[18]= |  |
Use
DSolve to obtain a symbolic solution to the PDE
,
:
| Out[10]= |  |
Use
NDSolve to obtain a numeric solution to the PDE
,
,
. The solution is stored as
for later use:
| Out[14]= |  |
Use
Plot3D to visualize the result of
NDSolve:
| Out[19]= |  |
In the cases above, the derivatives and partial derivatives are given in terms of variables. Here, they are evaluated numerically.
Both of the following ways can be used to express the values of partial differentials, such as
:
| Out[29]= |  |
| Out[28]= |  |
In addition to ODEs and PDEs,
Mathematica can also solve DAEs (differential-algebraic equations).
As an example, consider the DAE
,
satisfying
:
Use
DSolve to obtain a symbolic result for the DAE:
| Out[32]= |  |
To see the solution at
and
, use
to substitute these values into the solution:
| Out[33]= |  |
| Out[34]= |  |
You can also produce the same plot by first substituting
and
to
, and then using
NDSolve:
| Out[35]= |  |
As before, visualize the solution with
Plot:
| Out[36]= |  |
