# Ordinary Differential Equations (ODEs)

## Introduction

### Implicit and explit differential equations

An ordinary differential equation is one with only one independent variable. For example:

$$\dfrac{dy}{dx}=f(x)$$

The order of a differential equation is the number of differentials of $$y$$ included. For example one with the second derivative of $$y$$ is of order $$2$$.

Ordinary equations can can either implicit or explicit. An explicit function shows the highest order derivative as a function of other terms.

An implicit function is one which is not explicit.

A linear ODE is an explicit ODE where the derivative terms of $$y$$ do not multiply together, that is, in the form:

$$y^{(n)}=\sum_ia_i(x)y^{(i)}+r(x)$$

#### First-order ODEs

We have an evolution:

$$\dfrac{dy}{dt}=f(t,y)$$

And a starting condition:

$$y_0=f(t_0)$$

We now discuss various ways to solve these.

## First-order Ordinary Differential Equations

### Ordinary differential equations

An ordinary differential equation is one with only one independent variable. For example:

$$\dfrac{dy}{dx}=f(x)$$

The order of a differential equation is the number of differentials of $$y$$ included. For example one with the second derivative of $$y$$ is of order $$2$$.

Ordinary equations can can either implicit or explicit. An explicit function shows the highest order derivative as a function of other terms.

An implicit function is one which is not explicit.

A linear ODE is an explicit ODE where the derivative terms of $$y$$ do not multiply together, that is, in the form:

$$y^{(n)}=\sum_ia_i(x)y^{(i)}+r(x)$$

#### First-order ODEs

We have an evolution:

$$\dfrac{dy}{dt}=f(t,y)$$

And a starting condition:

$$y_0=f(t_0)$$

We now discuss various ways to solve these.

### Linear first-order Ordinary Differential Equations

#### Linear ODEs

For some we can write:

$$\dfrac{dy}{dt}=f(t,y)$$

$$\dfrac{dy}{dt}=q(t)-p(t)y$$

This can be solved by multiplying by an unknown function $$\mu (t)$$:

$$\dfrac{dy}{dt}+p(t)y=q(t)$$

$$\mu (t)[\dfrac{dy}{dt}+p(t)y]=\mu (t)q(t)$$

We can then set $$\mu(t)=e^{\int p(t)dt}$$. This means that $$\dfrac{d\mu }{dt}=p(t)u(t)$$

$$\dfrac{d}{dt}[\mu(t)y]=\mu (t)q(t)$$

$$\mu(t)y=\int \mu (t)q(t)dt + C$$

In some cases, this can then be solved.

#### Example

$$\dfrac{\delta y}{\delta x}=cy$$

$$y=Ae^{c(y+a)}$$

$$\dfrac{\delta^2 y}{\delta x^2}=cy$$

$$y=Ae^{\sqrt c (y+a)}$$

### Separable first-order Ordinary Differential Equations

For some we can write:

$$\dfrac{dy}{dt}=f(t,y)$$

$$\dfrac{dy}{dt}=\dfrac{g(t)}{h(y)}$$

We can then do the following:

$$h(y)\dfrac{dy}{dt}=g(t)$$

$$\int h(y)\dfrac{dy}{dt}dt=\int g(t)dt + C$$

$$\int h(y)dy=\int g(t)dt + C$$

In some cases, these functions can then be integrated and solved.

## Second-order Ordinary Differential Equations

### Linear second-order Ordinary Differential Equations

These are of the form

$$\dfrac{d^2y}{dt^2}+p(t)\dfrac{dy}{dt}+q(t)y=g(t)$$

There are two types. Homogenous equations are where $$g(t)=0$$. Otherwise they are heterogenous.

We explore the case with constants:

$$a\dfrac{d^2y}{dt^2}+b\dfrac{dy}{dt}+cy=0$$