**I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.**

## Week 2

In week 2 we began to look at homogenous second order linear differential equations (with constant coefficients). These are differential equations of the form

, (*)

where are real constants and the solution is . We showed that to solve these equations we look at the auxiliary equation

.

### Fact 1

*In general, if the roots of of this equation are and then the solution to (*) is given by*

*where and are real constants.*

The real constants are to be determined by two boundary/initial conditions of the form and .

There were possible complications however. If the roots are real and equal, say both , then the naive application of the above method yields:

,

which at the end of the day is just ‘constant’ .

A good question at this point is how do we know there are more solutions (as we are going to say below)? Before we even answer this we should note that the written here cannot satisfy a boundary condition of the form unless and this isn’t very interesting at all… i.e. .

We can do even better and design an actual damped harmonic oscillator/door closer such that the roots of the auxiliary equation are repeated. Now open the door to a width of 1 m and release from rest: — the initial speed is zero. Clearly something is missing and we can say this without going to deeply into the mathematics.

If we *do *go deeper into the mathematics we can go away and prove that the solution space of the linear operator:

is two dimensional (briefly, needs two independent components) so there must be another solution. As it turns out, and we did the calculation in class, will be a second solution.

### Fact 2

*If the roots of the auxiliary equation to (*) are repeated, then the general solution is given by*

*
*.

A second complication that can occur is if the roots are complex. The Conjugate Root Theorem says that the roots of the auxiliary equation will be of the form

whenever the coefficients are real (which they will be). In this case we *could *write the solution as

,

for some possibly complex constants and .

However it can be shown that this solution can be rewritten. This yields another fact:

### Fact 3

*If the roots of the auxiliary equation to (*) are complex, then the general solution is given by*

,

for real constants and .

## Week 3

In week 3 we asked what happens when the right hand side of (*) is not zero but rather some non-zero function of :

(**)

This is then a *non-homogeneous *second order linear differential equation. We were able to prove the following fact:

### N-HSOLDE Fact

*The general solution of (**) is given by*

,

*where is the solution to the associated homogeneous differential equation (*) and is a particular (i.e.* any) solution of (**).

The trick is to find a particular solution based on what . In general;

If , try . If , . If , . If , .

If , and or , . If it happens that does not satisfy (**) then it probably happens that is in fact a solution to (*). Try in this case.

## Tutorials

Tutorials Tuesday 6 – 8 pm start this week. If you have any questions this will be the time to ask. After that we will have tutorials in weeks 4, 5, 6, 8, 10, 12.

## Test 1

As it stands the test will be on in week 6 although I might be moving this to week 7. I will have a sample test for ye very soon.

## Notes

The notes on second order linear differential equations that I handed out. Note that you should be attempting the exercises.

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