MAT244-2013F > MidTerm

MT, P4

**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?

Navigation

[0] Message Index

[#] Next page

Go to full version