Class 08/23/2021

Differential Equations

These equations describe physical processes as mathematical models.

Decay of radioactive substances
Electrical networks
Population dynamics
Dispersion of pollutants
Trajectories of moving objects

Examples

Newtons Second Law of Motion

F = ma Write as diff eq for position f (t, y, \frac{d y}{d t}) = m \frac{d ^{2} y}{d t ^{2}} Position is y

Linear motion due to gravity

Let g denote constant acceleration due to gravity - approx $9.8 m / s^{2}$

w e i g h t = - m g m \frac{d ^{2} y}{d t ^{2}} = - m g because \frac{d ^{2} y}{d t ^{2}} is acceleration (mg=ma and mg=F) \frac{d ^{2} y}{d t ^{2}} = - g \frac{d y}{d t} = - g t + v_{0}, v_{0} is initial velocity at t = 0 y = - \frac{1}{2} g t^{2} = v_{0} t + y_{0}, y_{0} is initial position at t = 0

Radioactive Decay

\frac{d Q}{d t} = - k Q k > 0 proportionality constant. Q is the amount of material at time t Solution: Q (t) = Q_{0} e^{- k t}

ODE

More generally, $n^{t}$ th Order ordinary differential equations(ODE), are of the form

\frac{d ^{n}}{d ^{n}} = f (t, y, \frac{d y}{d t}, \dots, \frac{d ^{n - 1} y}{d t ^{n - 1}})

Ordinary since theres only one variable t
More thane variable yields partial differential equations (FDE), e.g. 1D wave equation $\frac{\partial ^{2} u}{\partial x ^{2}} = \frac{\partial ^{2} y}{\partial t ^{2}}$
Where u is a function of time $t$ And the spacial coordinate x

Initial Value Problems (IVP)

For example, we can show by substitution that $y = α e^{- 3 t} + 2 t + 1$ is the general solution to $\frac{d y}{d t} + 3 y = 6 t + 5$ . Subject to the initial condition $y (0) = 3$ Gives the particular where $3 = α + 1 = α = 2$