Differential Equation

An Equation in which the unknowns are functions. Solve for $y(t)$ in the below.

$$y^{\prime }(t)=3y(t)+t^3$$

Types

Ordinary Differential Equation Partial Differential Equation Linear Differential Equation Homogeneous Differential Equation Separable Differential Equation Systems of Differential Equations

Properties

Order

The order of a differential equation is the maximum derivative degree of the function of interest, e.g. $y(t)$, in the equation.

Techniques

Integrating Factor Separation of Variables

Special Examples

Population Dynamics

Linear Model

$$y^{\prime }=ry⟺y(t)=y_0{e}^{rt}$$

where $k$ is the net rate of change per capita

Nonlinear Logistic Model

Do not assume growth rate $g$ to be constant, but that it depends on $y$ Assume that when $y\approx 0,g\approx r>0$ Assume that when $y\approx k,g<0$ Where $r$ is the intrinsic rate Where $k$ is the carrying capacity

$$g=r\left(1-\frac{y}{k}\right)$$ $$y=\frac{K{y}_0}{y_0+(k-y_0)\exp (-rt)}$$

Newton’s Law of Cooling

$$T^{\prime }(t)=-k(T(t)-T_{\text{ambient}})⟺T(t)=(T_0-T_{\text{ambient}})e^{-kt}+T_{\text{ambient}}$$

Well Stirred Solution

todo

Basketball

todo

Maximum Interval of Existence

The maximal interval of existence refewrs to the largest interval over which a solution to an initial value problem (IVP) of a differential equation is defined and unique.

Case 1

For both Linear Differential Equation and Nonlinear Differential Equation, if $f(t,y)$ has discontinuities, solutions for $y(t)$ may not exist beyond the discontinuities of $f$, and the $t$ intervals of existence have become limited.

$$\frac{dy}{dt}=f(t,y)=\frac{t-y}{t-7}$$

General Solution

$$y(t)=\frac{t^2+C}{2(t-7)}$$

(t,y(t))	Maximal Interval of Existence
(-6,2)	$(-\infty ,7)$
(11,10)	$(7,\infty )$
Basically, the given point determines $C$, and depending on what $C$ is, the [[#maximum-interval-of-existence	Maximum Interval of Existence]] of the solution changes.

Case 2

For some Nonlinear Differential Equation, even if $f(t,y)$ is differentiable everywhere, solutions for $y(t)$ may not exist beyond some limited $t$ intervals.

$$\frac{dy}{dt}=\frac{y^2}{6}$$ $$y(t)=\frac{-6}{t+C}$$

The Maximum Interval of Existence follows the same pattern here. The point of case 2 is that even if your starting equation is continuous everywhere, your solution may not be, thus limiting the domain of $t$