ODEs

Python comes to great use when solving ODEs.

Solving ODEs

ODEs can be solved using the amazing sympy module and the dsolve function.

Here is how this can be done, step by step to solve the ODE $\frac{d^2y}{ {dx}^2}-3\frac{dy}{dx}+2y=0$:

Import sympy
```
 from sympy import *
```
Create symbols

We need to tell sympy that the letter x represents a symbol whilst y will be used to define the function.
```
 x = Symbol('x')
 y = Function('y')
```
Calculate derivatives

In sympy we can use the Derivative(y(x), x, i) function to calculate the ith derivative of the function y(x) with respect to x.
```
 y_ = Derivative(y(x), x)
 y__ = Derivative(y(x), x, 2)
```
Solve the ODEs

We can now solve the ODE using dsolve(eq, func) which will solve the equation eq for the function func.
```
 sol = dsolve(y__ - 3*y_ + 2*y(x), y(x))
```
Show the result

You can show a pretty formatted result using this command:
```
 pprint(sol)
```

Solving ODEs with initial conditions

We can also solve apply initial conditions to the dsolve function using the ics parameter.

For instance, to solve the ODE $3y”(x)+3y’(x)+4y(x)=0$ where $y(0)=3$ and $y’(0)=0$, we would apply the following ics: {y(0): 3, y(x).diff(x).subs(x, 0): 0}.

y(0): 3 defines the y(0)=3 condition
y(x).diff(x).subs(x, 0): 0} defines the y’(0)=0 condition. We first calculate the derivative of y(x) using diff(x), then we substitute x for the actual value which is 0.

Here is the full code:

from sympy import *

x = Symbol('x')
y = Function('y')

y_ = Derivative(y(x), x)
y__ = Derivative(y(x), x, 2)

sol = dsolve(3*y__ + 3*y_ + 4*y(x), y(x), ics={y(0): 3, y(x).diff(x).subs(x, 0): 0})

pprint(sol) # pprint formats the solution in a more readable format