MAT244-2018S > Final Exam

FE-P6

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Victor Ivrii:
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = -2xy\, , \\
&y'_t = x^2+y^2-1
\end{aligned}\right.
\end{equation*}

a. Linearize the system at
stationary points and sketch the phase portrait of this linear system.

b.  Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.

c. Sketch the full phase portrait.

Tim Mengzhe Geng:
For part(b), we have
\begin{equation}
(x^2+y^2-1)dx+2xydy=0
\end{equation}
Note that
\begin{equation}
M_y=N_x=2y
\end{equation}
The equation is exact.
By integration
\begin{equation}
H=\frac{1}{3}x^3+xy^2-x+h^\prime(y)
\end{equation}
\begin{equation}
h^\prime(y)=0
\end{equation}
We choose
\begin{equation}
h(y)=0
\end{equation}
In this way,
\begin{equation}
H(x,y)=\frac{1}{3}x^3+xy^2-x=C
\end{equation}
I will post solution to other parts later if no one else follows.

Nikola Elez:
I have attached a phase portrait

Syed Hasnain:
there is a small mistake..... in step 5 you have mentioned that h(y) = 0
it is notzero, it is a constant

Nikola Elez:
For part a)
Sorry if poor quality