Differential Equations (1 Viewer)

[leehuan]

I'm only an average 4u student right now. Bear with my stupidity.

So since I'm too dumb to interpret articles, can someone please explain how to know what to sub to find a differential equation's solution? This is 100% personal interest.

 

[anomalousdecay]

I'm not good at explaining proofs but imagine a case like this for 2nd order DEs:



Where f is some function.

Now with a differential equation, imagine we have f as a function of x in terms of exponentials, say

What happens if we differentiate this? Well,

and



So as you can see here, if we let the equation above at the top (but letting equal to zero), then we can obtain the homogeneous solutions to,



so as you can see by substituting these in, we get



What do we have here? A quadratic! Use the quadratic formula.

You will obtain two solutions at max (quadratic).

These solutions provide the necessary values of k as required.

Now, what we notice is that these values provide a solution in terms of the coefficients. However, if we had more than one solution, then we need to take into account both solutions. Say we get values . These are just solutions to the quadratic.

So what we have here is the roots to a random quadratic. But really, what is the quadratic we speak of?

One way to think of it is:

and

So,

So we have to take into account both of these. Do note however that here the coefficients of the quadratics could be anything as they are just constants. When taking this back to the function f, the constants can have any arbitrary value.

Hence to obtain the answer to f we have,



is for the particular solution by the way.

EDIT: You can find the values for R, Q and U by just using initial conditions for the f. For example, if f(0) = 0 and if f'(0) = 2, f''(0) = 1, then you can find these coefficients.

If there is a multiple root where , there is a special case where we have something else.

A much better explanation than what I have above can be found in Paul's online notes if you can find it. For the multiple root case, consult this:

http://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx


Say something if this doesn't make sense. I'm not all that great with explaining this type of stuff because I just usually find answers and put in the equation.

 

[VBN2470]

There are actually ways to solve DEs (Differential Equations), 3U/4U goes only as far as substituting a given expression to show that it is satisfies the equation etc. To know the solution to a DE depends on the type of DE and you won't really need to know this at the high school level. Your question is equivalent to "Can anyone tell me what substitution we need to use to solve an integral?", there is no one fixed substitution, there are many integrals with a different types of substitutions that you can use to solve the integral, it will depend on the type of integral you are being asked to evaluate, same case with DEs. Perhaps you should specify an example and maybe someone can direct you as to where you should be heading. If you're interested, check out: https://en.wikipedia.org/wiki/Ordinary_differential_equation and https://en.wikipedia.org/wiki/Examp...e_first-order_ordinary_differential_equations.

 

[anomalousdecay]

Also these are very useful in areas of science and engineering. For example, you can determine damping conditions from DEs, stability of systems, a systems general behaviour at different points in time, etc.

Also, a very powerful tool which you may come across in later years depending on your path is the Laplace transform, which can help you solve all sorts of DEs and generally become very helpful for the nth order case.

 

[Carrotsticks]

I'm only an average 4u student right now. Bear with my stupidity.

So since I'm too dumb to interpret articles, can someone please explain how to know what to sub to find a differential equation's solution? This is 100% personal interest.

This is an incredibly broad question. The substitution depends highly on the type of DE. What order is it? Is it non-linear? Homogenous? Ordinary?

A 4U equivalent to how broad your question is, would be "How do you find the substitution to solve an integral?"

 

[InteGrand]

You actually do often need to solve DE's in 4U Mechanics Q's (resisted motion), using things like etc. You can also use these results to derive the equation for for SHM from , or find a pair of functions who are derivatives of each other.

 

[leehuan]

Probably the first thing I need to be able to sort out is what type of DE a question could be. anomalousdecay I did kind of see where things were coming at with your example.

You actually do often need to solve DE's in 4U Mechanics Q's (resisted motion), using things like etc. You can also use these results to derive the equation for for SHM from , or find a pair of functions who are derivatives of each other.

Oh yeah, I've actually had to do SHM once and I completely didn't realise I was. Except I was using stuff like d/dx (v^2/2) and dy/dx=1/(dx/dy) which is all 3U have.
4U mechanics probably just sparked doubt in me, because there were always initial conditions.

 

[seanieg89]

It's very similar to the logical processes in choosing substitutions for indefinite integration. After all, integrating f is nothing more than solving the differential equation:

u' = f

for u.

You should first learn learn to recognise basic forms, and ways of classifying these things. (Such a systematic classification is pretty much impossible for PDE, but ODEs are much simpler.)
Ask yourself questions such as:
-Is the ODE linear?
-Are the coefficients constant?
-Is the equation homogeneous?
-What order is the ODE?

And know what the answers to each of these questions imply about how to go about solving your equation (or system of equations).

There are MANY books on this pitched at various levels of sophistication.

A final remark: you shouldn't expect an arbitrary ODE you write down to have a solution expressible in elementary functions, just like a typical function does not have elementary primitive.