# Differential Equations Solution Guide

A Differential Equation is an equation with a function and one or more of its derivatives:

Example: an equation with the function **y** and its derivative ** \frac{dy}{dx}**

In our world things change, and **describing how they change** often ends up as a Differential Equation.

Real world examples where Differential Equations are used include population growth, electrodynamics, heat flow, planetary movement, economic systems and much more!

## Solving

A Differential Equation can be a very natural way of describing something.

### Example: Population Growth

This short equation says that a population "N" increases (at any instant) as the growth rate times the population at that instant:

\frac{dN}{dt} = rN

But it is not very useful as it is.

We need to
**solve** it!

We **solve** it when we discover **the function** **y** (or
set of functions y) that satisfies the equation, and then it can be used successfully.

### Example: continued

Our example is **solved** with this equation:

N(t) = N_{0}e^{rt}

What does it say? Let's use it to see:

With **t** in months, a population that starts at 1000 (**N _{0}**) and a growth rate of 10% per month (

**r**) we get:

- N(1 month) = 1000e
^{0.1x1}=**1105** - N(6 months) = 1000e
^{0.1x6}=**1822** - etc

There is **no magic way to solve** all Differential Equations.

But over the millennia great minds have been building on each others work and have discovered different methods (possibly long and complicated methods!) of solving **some** types of Differential Equations.

So let’s take a
look at some different **types of Differential Equations** and how to solve them:

## Separation of Variables

Separation of Variables can be used when:

- All the y terms (including dy) can be moved to one side of the equation, and
- All the x terms (including dx) to the other side.

If that is the case, we can then integrate and simplify to get the the solution.

## First Order Linear

First Order Linear Differential Equations are of this type:

**P(x)**and

**Q(x)**are functions of x.

They are "First Order" when there is only ** \frac{dy}{dx}** (not ** \frac{d^{2}y}{dx^{2}}** or ** \frac{d^{3}y}{dx^{3}}** , etc.)

Note: a **non-linear** differential equation is often hard to solve, but we can sometimes approximate it with a linear differential equation to
find an easier solution.

## Homogeneous Equations

Homogeneous Differential Equations look like this:

v = \frac{y}{x}

which can then be solved using Separation of Variables .

## Bernoulli Equation

Bernoull Equations are of this general form:

\frac{dy}{dx} + P(x)y = Q(x)y^{n}

where n is any Real Number but not 0 or 1

- When n = 0 the equation can be solved as a First Order Linear Differential Equation.
- When n = 1 the equation can be solved using Separation of Variables.

For other values of n we can solve it by substituting u = y^{1−n} and turning it into a linear differential equation (and then solve that).

## Second Order Equation

Second Order (homogeneous) are of the type:

Notice there is a second derivative *d ^{2}y* dx

^{2}

The
** general** second order equation looks like this

a(x)*d ^{2}y* dx

^{2}+ b(x)

*dy*dx + c(x)y = Q(x)

There are many distinctive cases among these equations.

They are classified as homogeneous (Q(x)=0), non-homogeneous, autonomous, constant coefficients, undetermined coefficients etc.

For **non-homogeneous** equations the **general
solution** is the sum of:

- the solution to the corresponding homogeneous equation, and
- the particular solution of the non-homogeneous equation

## Undetermined Coefficients

The Undetermined Coefficients method works for a non-homogeneous equation like this:

\frac{d^{2}y}{dx^{2}} + P(x)\frac{dy}{dx} + Q(x)y = f(x)

where f(x) is a **polynomial, exponential, sine, cosine or a linear combination of those**. (For a more general version see Variation of Parameters below)

**guess**!

## Variation of Parameters

Variation
of Parameters is a little messier but works on a wider range of functions than the previous **Undetermined
Coefficients**.

## Exact Equations and Integrating Factors

Exact Equations and Integrating Factors can be used for a first-order differential equation like this:

M(x, y)dx + N(x, y)dy = 0

that must have some special function I(x, y) whose partial derivatives can be put in place of M and N like this:

\frac{∂I}{∂x}dx + \frac{∂I}{∂y}dy = 0

## Ordinary Differential Equations (ODEs) vs Partial Differential Equations (PDEs)

All of the methods so far are known as **Ordinary Differential Equations** (ODE's).

The term **ordinary** is used in contrast with the term *partial* to indicate derivatives with respect to only one independent variable.

Differential Equations with unknown multi-variable functions and their partial derivatives are a different type and require separate methods to solve them.

They are called **Partial Differential Equations** (PDE's), and
sorry, but we don't have any page on this topic yet.