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\centerline{\bf Ordinary Differential Equations\hfil Homework 1\hfil Childress, Spring 2002}
\centerline{Due January 29}
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1.  The {\it logistic} equation
 ${dy\over dx} = r y (k-y)$ is often used as a a model for population growth. Here
$r,k$ are positive contants, and we take $y \geq 0$. 
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(a) Sketch the direction field for the equation, indicating the form of integral
curves intersecting the $y$-axis both above and below $y=k$.

(b) Integrate to find the general solution of the initial value problem $y(0)= y_0> 0$.
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2. The {\it Ricatti} equation has the general form ${dy\over dx}= a(x) y^2+b(x) y + c(x)$.
This equation cannot in general be integrated. However it can be reduced to a linear
second-order equation. (a) Show that this is possible by the substitution 
$y= -a^{-1} u^{-1}{du/dx}$ and find the form of the equation. Find the solution of
the resulting equation when $a=e^x, b=x, c=0$ and $y(0)=y_0$.

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3. The equation of motion of a point $\rv(t) = (x(t),y(t),z(t))$ in rotation with fixed angular
velocity ${\bf \omega}= (\omega_1, \omega_2,\omega_3)$ satisfied the third-order linear system
$${d \rv\over dt}= {\bf \omega}\times \rv . \eqno(1)$$

(a) Show that for any solution $\rv(t)$ of (1) necessarily $ x^2+y^2+z^2 = R^2 =$ constant.

(b) Show that the transformation $\lambda = {x+iy\over R-z}$  (Stereographic projection) from the
sphere $ x^2+y^2+z^2 = R^2 $ to the complex $\lambda$ plane maps (1) into a 
complex valued Ricatti equation for $\lambda(t)$.
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4. Discuss the existence and uniqueness of solutions of the initial value problem
$${dy\over dx} = y^{1/2} + y^{3/2}, \;\; y(0) = y_0 \geq 0.$$
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