# 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.

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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 $\frac{dy}{dx}=y$.

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var( 'x' )
y = Function('y')(x)
dydx = Derivative( y, x )
ode = dydx - y
ode


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

You can solve an ODE by using the dsolve command.

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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.

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


$\displaystyle 5 = C_{1} e^{3}$

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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').

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


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

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