How to solve an ordinary differential equation

Description

Elsewhere we’ve seen how to write an ordinary differential equation. Once one is written, how can we ask software to solve it? And since ODEs often come with initial conditions that impact the solution, how can we include those as well?

Using SymPy, in Python

View this solution alone.

This answer assumes you have imported SymPy as follows.

1
2
Copy
from sympy import *                   # load all math functions
init_printing( use_latex='mathjax' )  # use pretty math output

Let’s re-use here the code from how to write an ordinary differential equation, to write $\frac{dy}{dx}=y$.

1
2
3
4
5
Copy
var( 'x' )
y = Function('y')(x)
dydx = Derivative( y, x )
ode = dydx - y
ode

${\displaystyle -y\left(x\right)+\frac{d}{dx}y\left(x\right)}$

You can solve an ODE by using the dsolve command.

1
2
Copy
solution = dsolve( ode )
solution

${\displaystyle y\left(x\right)=C_1{e}^x}$

If there are initial conditions that need to be substituted in for $x$ and $y$, it is crucial to substitute for $y$ first and then $x$. Let’s assume we have the initial condition $(3,5)$. We might proceed as follows.

1
2
Copy
with_inits = solution.subs( y, 5 ).subs( x, 3 )
with_inits

${\displaystyle 5=C_1{e}^3}$

1
Copy
solve( with_inits )

${\displaystyle \left[\frac{5}{e^3}\right]}$

To substitute $C_1=\frac{5}{e^3}$ into the solution, note that $C_1$ is written as var('C1').

1
Copy
solution.subs( var('C1'), 5/E**3 )

${\displaystyle y\left(x\right)=\frac{5e^x}{e^3}}$

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

Topics that include this task

• Bentley University GR526

Opportunities

This website does not yet contain a solution for this task in any of the following software packages.

• R
• Excel
• Julia

If you can contribute a solution using any of these pieces of software, see our Contributing page for how to help extend this website.