# How to write an ordinary differential equation

## Description

Differential equations are equations that contain differentials like $dy$ and $dx$, often in the form $\frac{dy}{dx}$. How can we write them using software?

## Using SymPy, in Python

This answer assumes you have imported SymPy as follows.

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

The following code tells SymPy that $x$ is a variable and that $y$ is a function of $x$. It then expresses $\frac{dy}{dx}$ as the derivative of $y$ with respect to $x$.

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var( 'x' ) # Let x be a variable.
y = Function('y')(x) # Literally, y is a function, named y, based on x.
dydx = Derivative( y, x ) # How to write dy/dx.
dydx # Let's see how SymPy displays dy/dx.

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

Letâ€™s now write a very simple differential equation, $\frac{dy}{dx}=y$.

As with how to do implicit differentiation, SymPy expects us to move everything to the left hand side of the equation. In this case, that makes the equation $\frac{dy}{dx}-y=0$, and we will use just the left-hand side to express our ODE.

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ode = dydx - y
ode

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

Content last modified on 24 July 2023.

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