- #1

psie

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- Homework Statement
- Consider the DE ##\frac{\mathrm{d}x}{\mathrm{d}t}=t\tan x## with initial value ##x(0)=\frac\pi{6}##.

1. Find the solution.

2. Describe the region in which the solutions are defined.

- Relevant Equations
- A separable first order ODE is of the form ##x'=g(x)h(t)##.

By inspection, we see that ##x=k\pi## is a solution for ##k\in\mathbb Z##. Moreover, the equation implicitly assumes ##x\neq n\pi/2## for odd ##n\in\mathbb Z##, since ##\tan x## isn't defined there. So suppose ##x\neq k\pi##, i.e. ##\tan x\neq 0##, then rearranging and writing ##\tan x=\frac{\sin x}{\cos x}## we have $$\frac{x'\cos x}{\sin x}=\frac{\mathrm{d}}{\mathrm{d}t}\log(|\sin x|)=t.$$ Integrating and exponentiating, we obtain, $$|\sin x|=Ae^{\frac{t^2}{2}}$$ The initial condition implies ##A=1/2##. Now, how do I get rid of the absolute values here? Does it somehow follow from the initial condition that ##\sin x## has to be positive?