# Ordinary Differential Equations

In this lesson, we will learn about solving first-order and higher-order ordinary differential equations.

## We'll cover the following

One of the most useful functions in modern problem-solving is that the solutions to ordinary differential equations (ODE) have so many varied applications in everything from pure sciences to all sorts of engineering.

$\frac{dy}{dx}+5yx^2=0$

# Tools

To set up the ODE, we need to refer to the function we are solving and its derivatives. The `Function`

and `Eq`

classes in SymPy help us do this. Let’s discuss these before we move on to solving some equations ourselves.

## Function

`Function`

is similar to `Symbol`

but is used to define a function.

```
f = Function('f')
```

To define a function of symbol `x`

, we use the following syntax:

```
x = Symbol('x')
f = Function('f')(x)
```

To compute the derivate of the `Function`

, we use the `diff()`

method:

```
f.diff(x) # first order derivative
f.diff(x, x) # second order derivative
```

## Equality

`Eq`

is used to define equations in SymPy.

```
Eq(rhs, lhs)
```

The comma separates the right-hand side from the left-hand side.

Now let’s use these tools to solve ODEs.

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