Lecture One Examples

Example 1

A population is growing at a rate that is proportional to it’s size

• Let $P(t)$ be the population at time $t$

$$P(t)=aP(t)$$

Rewrite $P(t)\to y$

$$y^{\prime }=ay$$

Divide both sides by $y$

$$\frac{y^{\prime }}{y}=a$$

Split fraction; Rewrite $y^{\prime }\to \frac{dy}{dt}$

$$\frac{1}{y}{y}^{\prime }=a\to \frac{1}{y}\frac{dy}{dt}=a$$

Multiply both sides by $dt$

$$\frac{1}{y}dy=adt$$

Integrate

$$\int \frac{1}{y}dy=\int adt\to ln|y|=at+c$$

y is a population, you can’t have negative population, so abs is dropped

$$ln(y)=at+c$$

By $e^{ln(y)}=y$:

$$y=e^{at+c}$$

Example 4

Undamped Pendulum

$$a^{\prime \prime }=\frac{-LMg\sin (a)}{M{L}^2}$$ $$a^{\prime \prime }=\frac{-g}{L}sin(a)$$