MAT244-2018S > Final Exam

FE-P6

**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

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