# Trigonometric First Order DE

1. Jan 13, 2009

### Matuku

$$\frac{dx}{dt} = \cos(x+t)$$

I'm having real troubles with this; I tried a substitution of $$u=x+t$$ but it just ends up as,

$$t=\tan{u} + \csc{u} + C$$

And I can't see where to go from there. (the middle function is cosec; it didn't come out very clearly on my screen).

2. Jan 13, 2009

### Dick

I keep getting t=csc(u)-cot(u)+C. You could then put u=x+t. But I don't think you can go much further than that as far as explicitly solving for x(t). It looks like kind of a mess.

3. Jan 13, 2009

### Matuku

You're right, it is -cot(u) rather than tan(u); don't know where I got that from. But you don't believe there's anyway to get it into a form x(t) [or even t(x)]?

Last edited: Jan 13, 2009
4. Jan 13, 2009

### Dick

I don't think so. You can try expressing everything in terms of sin and cos and then apply the addition formulas and hope things magically sort out. But it looks to me like x and t are pretty thoroughly mixed up.

5. Jan 13, 2009

### Staff: Mentor

That was my thought, too, looking at dx/dt = cos(x + t). I tried the cosine sum formula but didn't find any magic there. cos(x + t) apparently has x and t inextricably twined.