How to Solve Non-Linear Second Order Differential Equations?

[Unto]

Differential Equations again :(

y'' - y' = 0

How would I go about solving this? All I know is that the equation is not linear..

 

[Mark44]

It is linear. The characteristic equation is r² - r = 0.

 

[Unto]

so r = +/- sqrt(r)??

What am I supposed to do with it?

 

[Unto]

EDIT

Wrong equation. The actual question is:

y'' - 2y = 0

Is this linear? And how to solve?

I don't know this because my lecturer sucks and she just reads notes, I don't have a clue.

 

[Mark44]

No, no, no! How do you usually solve a quadratic equation?

 

[Hurkyl]

Well, you have a book don't you? Surely your book lays out a way to solve differential equations like this. And even if not, surely your class notes do?

What do they say the definition of linear is here?

 

[Unto]

Well I'm sorry if I suck at maths. I think 90% of maths is complete rubbish but I have to do it anyway. I study Physics so I have to do this. I look at my notes, all I see is some jerk wanking off with an equation but not explaining why he is doing what he did.

And I'm going to magically understand this how?

Please help I am begging you I won't sleep tonight

 

[Unto]

And this equation is not linear because I have a second derivative :/

y'' = d^2y/dx^2

 

[Unto]

This is a quadriatic equation?

 

[Mark44]

EDIT

Wrong equation. The actual question is:

y'' - 2y = 0

Is this linear? And how to solve?

I don't know this because my lecturer sucks and she just reads notes, I don't have a clue.

This is a linear differential equation as well. The characteristic equation this time is r² - 2 = 0. If you can solve this equation, we can go from there. If you can't solve this equation, you're probably going to have a very difficult time in this class, especially now that you have shared with us your opinion about math.

 

[Mark44]

And this equation is not linear because I have a second derivative :/

y'' = d^2y/dx^2

What you have here is two ways of writing the same thing, similar to writing 2x = x + x or
y'~=~dy/dx.

 

[Mark44]

so r = +/- sqrt(r)??

What am I supposed to do with it?

When you solve and equation for a particular variable, you end up with a new equation with that variable on only one side, and what it is equal to on the other. You haven't solved for r in your equation above.

 

[Unto]

No, no, no! How do you usually solve a quadratic equation?

This is a linear differential equation as well. The characteristic equation this time is r² - 2 = 0. If you can solve this equation, we can go from there. If you can't solve this equation, you're probably going to have a very difficult time in this class, especially now that you have shared with us your opinion about math.

r = +/-sqrt(2)

 

[Mark44]

OK, now we're making progress. Now that you have the roots of the characteristic equation, the general solution will be all linear combinations of two functions: e^r₁t and e^r₂t, where r₁ and r₂ are the numbers you found.

By linear combinations, I mean y = Ae^r₁t + Be^r₂t. All you need to do is to put in the two numbers and you're done.

You can check that what you get is a solution by calculating y'' - 2y. You should get 0.

 

[Unto]

What would A and B be?

And If I want to check the solution, all I do is differentiate my y equation and equation to y''?

 

[Mark44]

A and B can be any real numbers. Yes differentiate y to get y', then differentiate again to get y''. It should be the case that y'' - 2y is identically equal to 0.

 

[Unto]

Ok I'm beginning to get this. Thank you.

 

[Mark44]

Glad to hear it. You're welcome.

 

[Hurkyl]

Well I'm sorry if I suck at maths.

I wasn't criticizing your math skills -- I was criticizing your research skills. You don't have to know everything off the top of your head -- a big part of being good at math (or any other subject) is, if you don't know a bit of information, knowing how to find that information.

I'm assuming you have a book, or at least a set of prepared notes. A book is a valuable resource! If nothing else, knowing where to find information in your book is something that will be very valuable after a year goes by and you forget the details of solving.


If you really don't have a book and prepared notes and all you have is the notes you take in class, well, then you're at a disadvantage.

 

[nobahar]

This is a linear differential equation as well. The characteristic equation this time is r² - 2 = 0.

Hey guys,
Apologies for jumping in randomly, but where did this come from? How did this arise from y''-2y = 0?

 

[Mark44]

The OP started off with y'' - y' = 0, then realized he had given the wrong problem, which was actually y'' - 2y = 0.