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Differential Equations and a few helpful ways to solve them

Dr. V. Senthilkumar

Assistant professor

CPA college,Bodi

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Differential Equations

In this class we will focus on solving ordinary differential equations that represent the physical processes we are interested in studying. With perhaps a few exceptions the most complicated differential equation we will look at will be second order, which means it will look something like

Sometimes you may be able to solve the equation by hand. Other times not. Here we will cover some useful methods to help you solve them as you need to.

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SOS MATH

Great website that has so much helpful stuff on it and is pretty accessible and easy to follow, unlike so many others.

http://sosmath.com/diffeq/diffeq.html

A lot of the differential equations we are interested are solved there. I have always found it useful as it helps me learn to solve these things by hand.

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Mathematica

As a student at Notre Dame you can download and have access to Mathematica, which is an incredibly powerful and amazing tool for solving all sorts of mathematical problems. If it cannot solve the differential equation we are looking at then it is unlikely you can either.

The specific command for solving a differential equation is
DSolve

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Example

Solve the following differential equation in Mathematica assuming that D(x) and λ(x) are constants.

Rearrange (we will assume D is constant, but may not always be)

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In Mathematica

Open a clean sheet
Define you equation

Type eq:=DD*C’’[x}==lambda*C[x]
Hit shift+return

Confirm it is correct

Type eq
Hit shift +Return

Solve the differential equation

Type Dsolve[eq,C,x]
Hit shift+return

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Now you try

Solve the following differential equation

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You can also specify boundary conditions

Solve

Subject to boundary conditions

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You try

Solve

Subject to

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Now you’re in good shape to solve pretty much any of the ordinary differential equations that you will face in this class
You will have a few of these for homework just to make sure that you feel comfortable with it all.

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Numerical Solutions

Coupled Ordinary Differential equations.
Imagine I have two (or more) concentrations that are coupled to each other such that

Plot the concentrations over 100 times units

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Before we solve this

Look at the equations carefully.

What do you think the solution is going to look like

Do a few things

What is Steady State/Equilibrium
Where do both start, where do they end?
Draw what you think solution will look like.

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Hopefully you remember from JJW

You can approximate the derivative by (for small Δt)

So I can write my equations as

With these I can

write a loop to solve for

C1 and C2 at whatever

time I want

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Verify this solution

With Mathematica you can easily show that the theoretical solution to this problem is

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Ok you try this slightly harder one

Solve

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Alternative

If you prefer not to write your own solver and use a more efficient one you can use Matlab’s inbuilt ode45 function (https://www.mathworks.com/help/matlab/ref/ode45.html)
Or
You can use the numerical solver NDSolve in Mathematics (http://reference.wolfram.com/language/tutorial/NumericalSolutionOfDifferentialEquations.html) - I do not personally love this
On the following slide I show you how to solve some of the problems we have looked at with both, but leave it to you to invest time and learn how to use them.