Ordinary Differential Equations (ODEs)

Introduction

Order of differential equations

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:

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