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MTH61: DIFFERENTIAL EQUATIONS

This unit is about formation, types and solving differential equations.
MTH61: DIFFERENTIAL EQUATIONS 1

 Differential Equations

In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.

The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. Learn how to solve differential equations here.

One of the easiest ways to solve the differential equation is by using explicit formulas. In this article, let us discuss the definition, types, methods to solve the differential equation, order and degree of the differential equation, ordinary differential equations with real-word examples and a solved problem.

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

differential equation y + dy/dx = 5x
Example: an equation with the function y and its derivative dydx 

Solving

We solve it when we discover the function y (or set of functions y).

There are many “tricks” to solving Differential Equations (if they can be solved!).

But first: why?

Why Are Differential Equations Useful?

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

rabbits

Example: Rabbits!

The more rabbits we have the more baby rabbits we get.

Then those rabbits grow up and have babies too! The population will grow faster and faster.

The important parts of this are:

  • the population N at any time t
  • the growth rate r
  • the population’s rate of change dNdt

Think of dNdt as “how much the population changes as time changes, for any moment in time”.

Let us imagine the growth rate r is 0.01 new rabbits per week for every current rabbit.

When the population is 1000, the rate of change dNdt is then 1000×0.01 = 10 new rabbits per week.

But that is only true at a specific time, and doesn’t include that the population is constantly increasing. The bigger the population, the more new rabbits we get!

When the population is 2000 we get 2000×0.01 = 20 new rabbits per week, etc.

So it is better to say the rate of change (at any instant) is the growth rate times the population at that instant:

dNdt = rN

And that is a Differential Equation, because it has a function N(t) and its derivative.

And how powerful mathematics is! That short equation says “the rate of change of the population over time equals the growth rate times the population”.

Differential Equations can describe how populations change, how heat moves, how springs vibrate, how radioactive material decays and much more. They are a very natural way to describe many things in the universe.

What To Do With Them?

On its own, a Differential Equation is a wonderful way to express something, but is hard to use.

So we try to solve them by turning the Differential Equation into a simpler equation without the differential bits, so we can do calculations, make graphs, predict the future, and so on.

coin stack add

Example: Compound Interest

Money earns interest. The interest can be calculated at fixed times, such as yearly, monthly, etc. and added to the original amount.

This is called compound interest.

But when it is compounded continuously then at any time the interest gets added in proportion to the current value of the loan (or investment).

And as the loan grows it earns more interest.

Using t for time, r for the interest rate and V for the current value of the loan:

dVdt = rV

And here is a cool thing: it is the same as the equation we got with the Rabbits! It just has different letters. So mathematics shows us these two things behave the same.

Solving

The Differential Equation says it well, but is hard to use.

But don’t worry, it can be solved (using a special method called Separation of Variables) and results in:

V = Pert

Where P is the Principal (the original loan), and e is Euler’s Number.

So a continuously compounded loan of $1,000 for 2 years at an interest rate of 10% becomes:

V =1000 × e(2×0.1)
V =1000 × 1.22140…
V =$1,221.40 (to the nearest cent)

So Differential Equations are great at describing things, but need to be solved to be useful.

More Examples of Differential Equations

The Verhulst Equation

rabbits

Example: Rabbits Again!

Remember our growth Differential Equation:

dNdt = rN

Well, that growth can’t go on forever as they will soon run out of available food.

So let’s improve it by including:

  • the maximum population that the food can support k

A guy called Verhulst figured it all out and got this Differential Equation:

dNdt = rN(1−N/k)

The Verhulst Equation

Simple Harmonic Motion

In Physics, Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement. An example of this is given by a mass on a spring.

spring mass

Example: Spring and Weight

A spring gets a weight attached to it:

  • the weight gets pulled down due to gravity,
  • as the spring stretches its tension increases,
  • the weight slows down,
  • then the spring’s tension pulls it back up,
  • then it falls back down, up and down, again and again.

Describe this with mathematics!

The weight is pulled down by gravity, and we know from Newton’s Second Law that force equals mass times acceleration:

F = ma

And acceleration is the second derivative of position with respect to time, so:

F = m d2xdt2

The spring pulls it back up based on how stretched it is (k is the spring’s stiffness, and x is how stretched it is): F = -kx

The two forces are always equal:

m d2xdt2 = −kx

We have a differential equation!

It has a function x(t), and it’s second derivative d2xdt2

Note: we haven’t included “damping” (the slowing down of the bounces due to friction), which is a little more complicated, but you can play with it here (press play):

Creating a differential equation is the first major step. But we also need to solve it to discover how, for example, the spring bounces up and down over time.

Classify Before Trying To Solve

So how do we solve them?

walking

It isn’t always easy!

Over the years wise people have worked out special methods to solve some types of Differential Equations.

So we need to know what type of Differential Equation it is first.

It is like travel: different kinds of transport have solved how to get to certain places. Is it near, so we can just walk? Is there a road so we can take a car? Or is it in another galaxy and we just can’t get there yet?

So let us first classify the Differential Equation.

Ordinary or Partial

The first major grouping is:

  • “Ordinary Differential Equations” (ODEs) have a single independent variable (like y)
  • “Partial Differential Equations” (PDEs) have two or more independent variables.

We are learning about Ordinary Differential Equations here!

Order and Degree

Next we work out the Order and the Degree:

differential equation order 2, degree 3

Order

The Order is the highest derivative (is it a first derivative? a second derivative? etc):

Example:

dydx + y2 = 5x

It has only the first derivative dydx , so is “First Order”

Example:

d2ydx2 + xy = sin(x)

This has a second derivative d2ydx2 , so is “Order 2”

Example:

d3ydx3 + xdydx + y = ex

This has a third derivative d3ydx3 which outranks the dydx , so is “Order 3”

Degree

The degree is the exponent of the highest derivative.

Example:

(dydx)2 + y = 5x2

The highest derivative is just dy/dx, and it has an exponent of 2, so this is “Second Degree”

In fact it is a First Order Second Degree Ordinary Differential Equation

Example:

d3y/dx3 + (dy/dx)2 + y = 5x2

The highest derivative is d3y/dx3, but it has no exponent (well actually an exponent of 1 which is not shown), so this is “First Degree”.

(The exponent of 2 on dy/dx does not count, as it is not the highest derivative).

So it is a Third Order First Degree Ordinary Differential Equation

Differential Equations Solution Guide

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

differential equation y + dy/dx = 5x
Example: an equation with the function y and its derivative dydx 

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:

dNdt = 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) = N0ert

that can actually be used like this:

A population that starts at 1000 (N0) with a growth rate of 10% per month (r) will grow to

  • 1000e0.1×1 = 1105 in 1 month
  • 1000e0.1×6 = 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

Separation of Variables equations look like this:

dydx = xy

First Order Linear are of this type:

dydx + P(x)y = Q(x)

Homogeneous equations look like:

dydx = F ( yx )

Bernoulli are of this general form:

dydx + P(x)y = Q(x)yn, n ≠ 0 or 1

Second Order (homogeneous) are of the type:

d2ydx + P(x)dydx + Q(x)y = 0

Undetermined Coefficients and Variation of Parameters are both methods for solving second order equations when they are non-homogeneous like:

d2ydx + pdydx + qy = f(x)

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:

∂I∂xdx + ∂I∂ydy = 0

Separation of Variables

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

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First Order Linear

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

dydx + 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 dydx , not d2ydx2 or d3ydx3 , 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.

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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

  dydx = F ( yx )

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

v = yx

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

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Bernoulli Equation

A Bernoulli equation has this form:

dydx + P(x)y = Q(x)yn
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 = y1−n

    and turning it into a linear differential equation (and then solve that).

Find examples and read more about Bernoulli Equation

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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)d2ydx2 + b(x)dydx + 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

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Undetermined Coefficients

This method works for a non-homogeneous equation like

d2ydx2 + P(x)dydx + 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:

d2ydx2 + pdydx + 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:

    1. The general solution of the homogeneous equation

d2ydx2 + pdydx + qy = 0

    1. Particular solutions of the non-homogeneous equation

d2ydx2 + pdydx + 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

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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 y1 and y2

And when y1 and y2 are the two fundamental solutions of the homogeneous equation

d2ydx2 + pdydx + qy = 0

then the Wronskian W(y1, y2) is the determinant of the matrix

So

W(y1, y2) = y1y2‘ − y2y1‘

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

d2ydx2 + pdydx + qy = f(x)

using the formula:

yp(x) = −y1(x)∫y2(x)f(x)W(y1,y2)dx + y2(x)∫y1(x)f(x)W(y1,y2)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

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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:

∂I∂xdx + ∂I∂ydy = 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

Be careful not to confuse order with degree. Some people use the word order when they mean degree!

Linear

It is Linear when the variable (and its derivatives) has no exponent or other function put on it.

So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is).

More formally a Linear Differential Equation is in the form:

dydx + P(x)y = Q(x)

Solving

OK, we have classified our Differential Equation, the next step is solving.

And we have a Differential Equations Solution Guide to help you.

Assignment

DIFFERENTIAL EQUATIONS ASSIGNMENT

ASSIGNMENT : DIFFERENTIAL EQUATIONS ASSIGNMENT MARKS : 20  DURATION : 1 week, 3 days

 

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