Differential Equations

127. Differential Equations

A differential equation specifies a relationship between an unknown function and its derivatives

Solving for (an) unknown function(s) that satisfy this relationship

Initial or boundary conditions to select a unique solution

127.0.1. Linearity

The dependent variable and its derivatives each appear only to the first power, are not multiplied together, and are not inside any other function

Example

Linear

\frac{d 𝑦}{d 𝑡} + 𝑦 = 0

Non-Linear

\begin{matrix} \frac{d 𝑦}{d 𝑡} + 𝑦^{2} = 0 \\ \frac{d 𝑦}{d 𝑡} + sin (𝑦) = 0 \\ \frac{d 𝑦}{d 𝑡} + 𝑒^{𝑦} = 0 \end{matrix}

127.0.2. Homogeneity

Homogeneous: No terms with just the independent variable of constant
Nonhomogeneous: Has term(s) with just the independent variable or constants

Example

Homogeneous

\frac{d 𝑦}{d 𝑡} + 𝑦 = 0

Non-Homogeneous

\begin{matrix} \frac{d 𝑦}{d 𝑡} + 𝑦 = 1 \\ \frac{d 𝑦}{d 𝑡} + 𝑦 = sin (𝑡) \\ \frac{d 𝑦}{d 𝑡} + 𝑦 = 𝑒^{𝑡} \end{matrix}

127.0.3. Order & Degree

Order: highest derivative
Degree: exponent on highest derivative

Example

Order

\begin{aligned} \frac{d 𝑦}{d 𝑥} + 𝑦 = 𝑥 & 1st order \\ \frac{d^{2} 𝑦}{d 𝑥^{2}} + 𝑦 = 𝑥 & 2nd order \end{aligned}

Degree

{(\frac{d^{2} 𝑦}{d 𝑥^{2}})}^{3} + 𝑦 = 𝑥 3rd degree

127.0.4. Autonomous

The independent variable does not appear explicitly in the equation (rate of change depends only on the dependent variable)

Example

Autonomous

\frac{d 𝑦}{d 𝑡} = 𝑦

Non-Autonomous

\frac{d 𝑦}{d 𝑡} = 𝑦 + 𝑡

127.0.5. Ordinary Differential Equation (ODE)

One independent variable
Involves ordinary derivatives with respect to that variable

127.0.6. Partial Differential Equation (PDE)

Multiple independent variables
Involves partial derivatives with respect to those variables

127.0.7. Summary

Inputs
- Equation itself
- Initial/boundary conditions (optional)
Outputs
- Function(s) that satisfy it
Defined for continuous variables

Example

Non-coupled ODE

Exponential Population Growth

\frac{d}{d 𝑡} 𝑃 (𝑡) = 𝑟 \cdot 𝑃 (𝑡)

$𝑃 (𝑡)$ : population at time $𝑡$
$\frac{d}{d 𝑡} 𝑃 (𝑡)$ : rate of change of the population at time $𝑡$
$𝑟$ : growth rate constant
$𝑟 \cdot 𝑃 (𝑡)$ : growth contribution proportional to the current population

Example

Coupled ODE

Lotka-Volterra ODEs (predator-prey)

\begin{aligned} \frac{d}{d 𝑡} 𝑥 (𝑡) & = 𝛼 𝑥 (𝑡) - 𝛽 𝑥 (𝑡) 𝑦 (𝑡) \\ \frac{d}{d 𝑡} 𝑦 (𝑡) & = 𝛿 𝑥 (𝑡) 𝑦 (𝑡) - 𝛾 𝑦 (𝑡) \end{aligned}

$𝑥 (𝑡)$ = prey population
$𝑦 (𝑡)$ = predator population
$\frac{d}{d 𝑡} 𝑥 (𝑡)$ : The rate of change of the prey population at time $𝑡$
$\frac{d}{d 𝑡} 𝑦 (𝑡)$ : The rate of change of the predator population at time $𝑡$
$𝛼$ : Prey growth rate — how fast the prey multiply in the absence of predators
$𝛽$ : Predation rate — how effectively predators consume prey
$𝛿$ : Predator growth rate — how much predator population increases when consuming prey
$𝛾$ : Predator death rate — how fast predators die without enough food
$𝛼 𝑥 (𝑡)$ : prey naturally reproduce at a rate proportional to how many exist now
$- 𝛽 𝑥 (𝑡) 𝑦 (𝑡)$ : prey lost to predation, proportional to encounters with predators
$𝛿 𝑥 (𝑡) 𝑦 (𝑡)$ : predators grow in number based on how much prey they consume
$- 𝛾 𝑦 (𝑡)$ : predators die naturally when there isn’t enough food

127.1. Slope Field (Direction Field)

Example

IVP (Initial Value Problem)

{\begin{cases} \frac{d 𝑦}{d 𝑥} = 𝑦 \\ 𝑦 (2) = 1 \end{cases}

Example

IVP (Initial Value Problem)

{\begin{cases} \frac{d 𝑦}{d 𝑥} = 𝑥 - 𝑦 \\ 𝑦 (−5) = 1 \end{cases}