E.D.O - Coefficient not constant

Question

Hello Guys i've been being difficulty to solve this Differential equation, because the coefficients aren't constant. I don't have an idea to solve.

$\frac{dx}{dt}*sin(t) + x*\cos(t) = 1 , (t,x) \in (0,\pi)\times\mathbb{R}$

I'd like to find a general solution.

Thanks.

score 0 · Answer 1 · answered Apr 07 '18 at 16:03

0

Note that $$\frac{dx}{dt}*sin(t) + x*\cos(t) = \frac {d}{dt} ( x\sin t ) $$

Thus you have $$\frac {d}{dt} ( x\sin t )=1 $$

Upon integrating you get

$$ x \sin t = t+c $$

Solve for x and find your constant from initial conditions.

answered Apr 07 '18 at 16:03

but you did the second equations because you realize or is there something to discover it ? – Robert Mota Apr 07 '18 at 16:06
Most of the times you need an integrating factor. In your problem the integrating factor is not needed because it is a total derivative by itself. I just used product rule on $ x\sin t$. – Mohammad Riazi-Kermani Apr 07 '18 at 16:20
Thanks so much man :D – Robert Mota Apr 07 '18 at 16:25

1 Answers1