Can You Solve This Nonlinear Differential Equation with Cosine Functions?

[latentcorpse]

ok so try this guy

$$\lambda (\frac{dx}{dt})^{2} = (cos{ \alpha} - x)(cos{ \beta} - x)$$
where $$x = cos{ \theta}$$

and the solution is

$$x = cos{ \alpha} sin^{2}{\frac{t}{2 \lambda}} + cos{ \beta} cos^{2}{\frac{t}{2 \lambda}}$$

please show you're working cos that's where I am getting lost

 

[mrandersdk]

maybe I am reading your post wrong, but I don't see any question?

 

[Integral]

Why not show us YOUR working, then maybe we can spot where you are going wrong.

 

[latentcorpse]

ok so the question is to solve the differential equation to try and prove that is indeed the solution.

however all I've managed to do so far is square root everything and crossmultiply - what would u reckoon the substituion is?

 

[mrandersdk]

why do you say that $$x = cos{ \theta}$$ and then say that the solution is $$x = cos{ \alpha} sin^{2}{\frac{t}{2 \lambda}} + cos{ \beta} cos^{2}{\frac{t}{2 \lambda}}$$, that doesn't make sense to me.

But it should be farely easy to check that it is a solution, just put x in on both sides and see if they agree.

 

[latentcorpse]

yeah god point i think that's a mistake

ok. forget the x = cos(theta) part

 

[mrandersdk]

maybe it is not, maybe I just misread it in my head, you can of cause find theta by

$$x = cos^{-1}[cos{ \alpha} sin^{2}{\frac{t}{2 \lambda}} + cos{ \beta} cos^{2}{\frac{t}{2 \lambda}}]$$

and that makes sense. But I don't understand why you just don't check it yourself? And if you can't get all the way through show us what you have done and we will try to correct you if it is wrong, or simply help you further

 

[ice109]

mathematica gives me a novella for an answer so if you have an analytic soln be clearer about it