FIRST ORDER EQUATIONS MODELING

Linear Equations

Radioactive Decay:

Q' = -rQ,
with decay rate r, and half life $\tau = ln(2)/r$.

Compound Interest:

S' = rS, interest rate r;
S' = rS + k, includes deposit or withdrawal rate k.

Mixing:

$Q' = r(\alpha-\beta Q)$.

Temperature after Death:

$\theta' = -k(\theta-T)$,
with ambient temperature T.

Mechanics Problems

come from
Newton's Law of Motion: F = ma = mv' = mx''.

• dropped mass with air resistance $\sim \vert v\vert$:
  mv' = mg - kv.
• mass moving away from earth:
  $mv' = -mg\frac{R^2}{(R+x)^2}$.

LINEAR FIRST ORDER THEORY

Solution Theorem:

if p and g are continuous on $I = (\alpha, \beta)$,
with $t_0 \in I$, then there exists a unique y(t) that satisfies
y' + p(t)y = g(t), for all $t \in I$ given any y₀ = y(t₀).
Note: if $\mu(t) = exp(\displaystyle\int_{t_0}^t p(t)dt)$, so that $\mu(t_0) = 1$, then

$$y(t) = (\ y_0 + \int_{t_0}^t\mu(t)g(t)dt\ )/\mu(t).$$

NONLINEAR FIRST ORDER THEORY

General Equation:

y' = f(t,y), given y(t₀) = y₀.

Solution Theorem:

if f and $\partial f/\partial y$ are continuous for
$D = \{ (y,t) : \alpha < t < \beta,\ \gamma < y < \delta\}$ , with $(t_0,y_0) \in D$,
then for some h>0 there exists a unique y(t) that satisfies y' = f(t,y), for all $t \in (t_0-h,t_0+h)$, with y₀ = y(t₀).

Issues

• Uniqueness? Example y' = y^1/3, with y(0)=0,
  has solution y=(2t/3)^3/2 or y=-(2t/3)^3/2.
• Interval of definition? Example y' = y², with y(0)=1,
  has solution y=1/(1-t); discontinuity at t = 1 cannot be anticipated from the original problem f(t,y) = y².
• Implicit Solutions: solutions are often found only in implicit form F(t,y) = 0 (the integral of the DE).