ODE with trigonometric function: xy' = y + 3x²cos²(y/x)

[asdf1]

for this O.D.E.
xy` = y + 3(x^2)[cos^2(y/x)], y(0)=pi/2
how do you work this out?

 

[GCT]

substitute y=vx

 

[Benny]

Divide both sides by x first. That will make it easier to see why GCT's substitution works.

 

[asdf1]

i think I'm having trouble with when are you supposed to make that kind of substitution...
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[TD]

Did you do what Benny said? That would give:

$$xy' = y + 3x^2 \cos ^2 \left( {\frac{y}{x}} \right) \Leftrightarrow y' = \frac{y}{x} + 3x\cos ^2 \left( {\frac{y}{x}} \right)$$

That should make it more clear to see the possible substitution of y/x.

 

[asdf1]

yes... but probably I'm not good with numbers, because i still wouldn't think of using that substitution in the first place... I'm not sharp~

 

[TD]

The y/x in the cos should ring a bell, as well as the factor x before the y'. To get the y' isolated, dividing by x makes another y/x of the first term on the RHS (the y).

It's a bit looking and trying

 

[Benny]

Ok if you find it to be slightly difficult to 'see' the kinds of manipulation are required then the following might help. If you've just started ODEs then it's unlikely that that you will be dealing with a large variety of DEs. The first few types that you usually learn are separable, first order linear and homogeneous (some textbooks call a whole bunch of different DEs homogeneous but don't worry too much about that if you're just starting out). So basically try to write the DE in the standard first order linear form. If you can't then try to separate variables. Can't do that? The chances are that it's homogeneous. It doesn't matter if it doesn't look homogeneous, just make the substitution y = vx and the algebra will sort itself out.

The above 'advice' only really helps if you've just started doing DEs. It tends to become a little impractical to deduce the form of the DE during an exam situation once you start learning more ODEs. This is especially true when the assignment/test/exam setter decides to be tricky and put in a question which is neither linear, separable or homogeneous. It happened on my last assignment. By the way, there's no need to be so humble about your abilities. These questions become quite easy (in general) once you've done a lot of them. A trap is to just watch someone else do the question and not do any yourself. The difference between my math and physics scores illustrates this point. ;)

Edit: Perhaps you could fill us in on the material that you've covered. Sometimes you might just come across questions that you can't do because you've never seen questions of that type before.

 

[asdf1]

hmm... you're right! I'm starting D.E's... I'm studying O.D.E's on my own, because i think it's a very important application in science~
right now, I'm looking at engineering mathematics by kreygzig, and having a little trouble trying to understand~
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[asdf1]

thank you very much! :)