Ordinary Differential Equations

ODE Basics

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01

Vector Field

Concept
Vector Field

A vector field assigns an arrow to every point $(x, t)$ in the plane. Each arrow shows the direction and speed that a solution passing through that point must travel.

Arrow At any point $(x, t)$, the arrow points in direction $(1,\, f(x,t))$: one unit forward in time, $f(x,t)$ units in $x$.
Length Arrow length encodes speed: long arrows mean $x$ changes rapidly; short arrows mean slow change.
Nullcline Where $f(x,t)=0$ the arrows are horizontal: $x$ is momentarily not changing.
Example
$\dot{x} = -x$
Exponential decay. Every solution converges to 0.
Interactive
Draw the field · click to trace a path
Vector Field · click to trace a solution
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↑ each arrow = the ODE's instruction at that point
02

General Solution

Concept
General Solution

The general solution is the complete family of all functions that satisfy the ODE. It contains a free constant $C$ that parameterises every possible trajectory.

Example   $\dot{x} = -x$
1
Separate variables: $\tfrac{dx}{x} = -\,dt$
2
Integrate both sides: $\ln|x| = -t + K$
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Exponentiate: $x(t) = C\,e^{-t}$, where $C = e^K$
$x(t) = C\,e^{-t}$
C free $C$ can be any real number. Each value gives a different curve. The general solution is not one curve; it is a whole family.
Degree An $n$-th order ODE typically has $n$ free constants in its general solution.
Interactive
Family of curves · vary C
General Solution   $x(t) = C\,e^{-t}$
1.0
highlighted = current C   grey = rest of the family
03

Particular Solution

Concept
Particular Solution

A particular solution is a single curve selected from the general solution by imposing an initial condition $x(t_0) = x_0$.

Example
1
Start with the general solution $x(t) = C\,e^{-t}$
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Plug in the initial condition: $x(0) = x_0 \Rightarrow C\,e^{0} = x_0$
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Solve for C: $C = x_0$. The particular solution is $x(t) = x_0\,e^{-t}$
$x(t) = x_0\,e^{-t}$
IVP ODE + initial condition = Initial Value Problem (IVP). The particular solution is the unique answer to an IVP.
Interactive
Set the initial condition $x(0) = x_0$
Particular Solution · IVP
2.00
red dot = initial condition   orange curve = particular solution
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Side by Side

Summary
General vs Particular Solution
General Solution Particular Solution
Contains Free constant $C$ No free constants
Represents All solutions (a family) One specific solution
Needs Only the ODE ODE + initial condition
Example $x(t) = C\,e^{-t}$ $x(t) = 2\,e^{-t}$
On graph Infinitely many curves A single curve through $(0, x_0)$
$\dot{x} = -x$ · family + particular
2.00