MAT244-2013F > MidTerm

MT, P4

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Victor Ivrii:
Analyze the direction field and constant (equilibrium) solutions of the ODE
\begin{equation*}
y'=\frac{\sin y}{1+\sin^2 t}
\end{equation*}
to explain why the solution $y(t)$ of the initial value problem
\begin{equation*}
y'=\frac{\sin y}{1+\sin^2 t},\qquad y(0)=1
\end{equation*}
is defined for all values of $t$, is an increasing function  and  satisfies the inequality $0<y(t)<\pi$ for all values of $t$.

(Do not try to solve the initial value problem.)

Xiaozeng Yu:
4

Xuewen Yang:
Just wondering, for this question, do we need to draw the direction field?

Victor Ivrii:
Solution is incorrect (it is based on presumption that $y$ takes values in $(0,\pi)$ instead of proving it).

Xiaozeng Yu:
ahh...omg, because the rectangle must contain the initial value point (0,1) in order to have an unique solution of the initial value problem. sin1>0 make the function increasing, so the retangle (a<0<b, 0<y<pi) which containing the initial point contains the unique solution function of (0,1) which is increasing?

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