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\centerline{\bf Ordinary Differential Equations\hfil Homework 4\hfil Childress, Spring 2002}
\centerline{Due February 19}
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1. Locate and classify all of the critical points of

$$ \quad\dot{x}=y, \;\dot{y}=-ay -b \sin{x}$$
as a function of the constants $a,b$, subject to the conditions
$a \geq 0, b >0$. If $a>0$, this system describes a simple nonlinear pendulum with
damping (with $x=\theta, y=\dot{\theta}$). Based on the intuition provided by this fact, if
$a > 0$,  what happens to orbits passing through points on the positive $y$ axis, as $t\rightarrow\infty$?
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2. The case $\dot{x}=Ax+By, \dot{y}=Cx+Dy$ with $AD=BC$ is singular or {\it degenerate} .
Investigate this case by finding equations for $Dx-By$ and $Ax+By$. Sketch orbits in 
in the phase plane for the case $A=1, B=2, C=3, D=6$.
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3. Consider the system $\dot{x}=-x-y^2,\; \dot{y}= x$. Classify the critical
point at the origin  and sketch orbits of the linearized system in the phase plane. Then assess the
effect of the nonlinear term and sketch the phase plane of the full system, by sketching in direction vectors or using any graphing aids you may have.
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4. Prove that the second-order equation
$${d^2 z\over dt^2}+(z^2+2({dz\over dt})^2-1){d z\over dt}+z=0$$
has a nontrivial periodic solution. First convert to a first-order system for $x(t)=z,y(t)={d z\over dt}$.(Hint: consider a suitably chosen annulus $r_1^2 < x^2+y^2 < r_2^2$.)
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5. Green's theorem in the plane states that, if $f(x,y), g(x,y)$ are continuous and have continuous derivatives
in a simple region $R$ bounded by a simple closed curve $C$, 
then $\oint_C [f(x,y)dy-g(x,y)]dx=\int\int_R (f_x+g_y)dxdy$. Suppose
$\dot{x}=f(x,y), \dot{y}=g(x,y)$, and that $C$ is a periodic orbit of this system. 

(a) Show that in this case the line integral above will vanish, and $f_x+g_y$ cannot have
one sign over $R$.

(b) Prove that the system $\dot{x}=x-xy^2+y^3, \dot{y}=3y-yx^2+x^3$ has no periodic solution within the
circle $x^2+y^2=4$.

(c) Show that this last result cannot be proved by examining the sign of $d(x^2+y^2)/dt$ on $x^2+y^2=4$.

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