# 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, economical systems and much more!

## Solving

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

### Example: Population Growth

Here we say 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}

that can actually be used like this:

A population that starts at 1000 (N_{0}) with a growth rate of 10% per month (r) will grow to

- 1000
_{}e^{0.1x1}= 1105 in 1 month - 1000
_{}e^{0.1x6}= 1822 in 6 months - etc

There is no magic bullet 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

^{n}, n ≠ 0 or 1

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

has 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

## 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, you will then have to integrate and simplify the solution.

Read more about Separation of Variables

## First Order Linear

A **first order differential equation** is **linear** when it
can be made to look like this:

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

Where **P(x)** and **Q(x)** are functions of x.

Observe that 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.

If you have an equation like this then you can read more on Solution of First Order Linear Differential Equations

**Note:**non-linear differential equations are often harder to solve and therefore commonly approximated by linear differential equations to find an easier solution.

## Homogeneous Equations

There is another special case where Separation of Variables can be used called homogeneous.

A first-order differential equation is said to be homogeneous if it can be written in the form

\frac{dy}{dx} = F ( \frac{y}{x} )

Such an equation can be solved by using the change of variables:

v = \frac{y}{x}

which transforms the equation into one that is separable. To discover more on this type of equations, check this complete guide on Homogeneous Differential Equations

## Bernoulli Equation

A Bernoulli equation has this 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 = yand turning it into a linear differential equation (and then solve that).
^{1−n}

Find examples and read more about Bernoulli Equation

## Second Order Equation

In this type of equation the second derivative makes its appearance. The general second order equation is written as follows 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 equal to the sum of:

*Solution to corresponding homogeneous
equation*

*+*

*Particular solution of the
non-homogeneous equation*

Find out more about these equations

## Undetermined Coefficients

This method works for a non-homogeneous equation like

\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.

To keep things simple, we only look at the case:

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

where **p** and **q** are constants.

The **complete solution** to such an equation can be found
by combining two types of solution:

- The
**general solution**of the homogeneous equation **Particular solutions**of the non-homogeneous equation

\frac{d^{2}y}{dx^{2}} + p\frac{dy}{dx} + qy = 0

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

Once we have found the general solution and all the particular solutions, then the final complete solution is found by adding all the solutions together.

This method also involves making a **guess**! Read more at Undetermined
Coefficients

## Variation of Parameters

This is a more general method than Undetermined Coefficients.

Once you have the general solution to the homogeneous equation, you
have two fundamental solutions y_{1} and y_{2}

And when y_{1} and y_{2} are the two fundamental
solutions of the homogeneous equation

\frac{d^{2}y}{dx^{2}} + p\frac{dy}{dx} + qy = 0

then the Wronskian W(y_{1}, y_{2}) is the determinant
of the matrix

So

W(y_{1}, y_{2}) = y_{1}y_{2}'
− y_{2}y_{1}'

And using the Wronskian we can now find the particular solution of the differential equation

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

using the formula:

y_{p}(x) = −y_{1}(x)∫\frac{y_{2}(x)f(x)}{W(y_{1},y_{2})}dx
+ y_{2}(x)∫\frac{y_{1}(x)f(x)}{W(y_{1},y_{2})}dx

Finally we complete solution by adding the general solution and
the particular solution together.

You can learn more on this at Variation of Parameters

## Exact Equations and Integrating Factors

An "exact" equation is where a first-order differential equation like this:

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

has 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

and our job is to find that magical function I(x,y) if it exists.

Find out how to solve these at Exact Equations and Integrating Factors

## 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.