The discussion revolves around solving the differential equation \( x' = \sec(x) \) with the initial condition \( x(0) = 2 \). Participants explore the implications of the solution \( \sin(x) = t + c \) and the constraints of not using the arcsin function to express \( x \). The conversation includes considerations of initial conditions and the behavior of the secant function.
Participants express differing views on the necessity of using the arcsin function to solve for \( x \). While some believe it is unavoidable, others explore alternative representations and interpretations of the solution. The discussion remains unresolved regarding the best approach to avoid arcsin while still addressing the initial value problem.
Participants highlight the importance of the initial condition and the behavior of the secant function, which may introduce complexities in the solution process. The discussion also reflects uncertainty about how to express the solution without using arcsin.
(i) What is the initial condition? x(0) = ??simo said:I was given an DE equa x'=sec(x) : x(0)
I then solved it to
sin(x)= t + c(c0nstant)
but then I can't solve for x cause they say we shouldn't use arcsin. what can I do
Opalg said:(i) What is the initial condition? x(0) = ??
(ii) There may be some trickery in this question. The sec function is undefined (or "infinite") at some points, so the equation needs to be interpreted cautiously at any such point.
In that case I don't see any way of avoiding an inverse sin function.simo said:it it x(0)=2
Opalg said:In that case I don't see any way of avoiding an inverse sin function.
You solution $\sin x = t +c$ is correct. You can find the constant from the initial condition. Then you can either leave the solution in that form, or you can write it as $x = \arcsin(t+c)$. There is no way of writing the arcsin function in terms of other functions.simo said:are you saying the only way possible to solve this is by the arcsin
Opalg said:You solution $\sin x = t +c$ is correct. You can find the constant from the initial condition. Then you can either leave the solution in that form, or you can write it as $x = \arcsin(t+c)$. There is no way of writing the arcsin function in terms of other functions.