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02. Fundamentals

Dated: 07-11-2024

Growth Equation

\[\frac{du}{dt} = F(t)G(u)\]

The Pendulum Equation

\[\frac{d^2\theta}{dt^2} + \frac g l \sin(\theta) = F(t)\]

The Van Da Pol Equation

\[\frac{d^2y}{dt^2} + \epsilon (y^2 + 1) \frac {dy}{dt} + y = 0\]

The Lcr Oscillator Equation

\[L \frac{d^2 Q}{dt^2} + R \frac{dQ}{dt} + \frac Q C = E(t)\]

A Riccati Equation

\[\frac {dp}{dt} = - 2a (t)p + \frac{b(t)^2}{u(t)}p^2 - v(t)\]

Ordinary Differential Equation

If an equation contains an ordinary derivative1 of one or more dependent variables then it is called ordinary differential equation.

Example

\[\frac{d^{2}y}{dx^{2}}+5\left(\frac{dy}{dx}\right)^{3}-4y=e^{x}\]

Partial Differential Equation

If an equation contains a partial derivative2 of one or more dependent variables then it is called partial differential equation.

\[a^2 \frac{\partial^4 u}{\partial x^4} + \frac{\partial^2 u}{\partial x^2} = 0\]

Imagine if the following function3 defined over an interval4 \(I\)

\[f\left(t, y, y^\prime, \ldots, y^{(n)}\right) = 0\]

such that

  1. \(y(t)\) and its first \(n\) derivatives1 exist over the interval4 \(I\) for all \(t\) and its first \(n - 1\) derivatives1 are continuous[^5] over \(I\).
  2. \(y(t)\) satisfied the differential equation for all \(t\) in \(I\).

Initial Value Problem Examples

The Logistic Equation

\[p^\prime = ap - bp^2\]

With initial condition \(p(t_0) = p_0\) for \(p_0 = 10\), the solution is

\[p(t) = \frac {10 a}{10 b + (a - 10b)e - a(t - t_0)}\]

The Mass Spring System Equation

\[x^{\prime\prime} + \left(\frac a m \right) x^\prime + \left(\frac k m\right)x = g + \left(\frac{F(t)}{m}\right)\]

Boundary Value Problem

Example 1

\[y^{\prime\prime} + 9y = \sin(t)\]

with initial conditions \(y(0) = 1, y^\prime(2p) = -1\), solution is

\[y(t) = \left(\frac 1 8\right)\sin(t) + \cos(3t) + \sin(3t)\]

Example 2

\[y^{\prime\prime} + p^2y = 0\]

With initial conditions \(y(0) = 2, y(1) = -2\), solution would be

\[y(t) = 2 \cos(pt) + (c) \sin (pt)\]

Properties

Linear

If the \(nth\) order differential equation can be written as following then it is called linear

\[a_n(t)y(n) + a_{n - 1}(t) y (n - 1) + \ldots + a_1y^\prime + a_0(t)y = h(t)\]

Non Linear

\[x_3(y^{\prime\prime\prime})3 - x_2y(y^{\prime\prime}) + 3xy^\prime + 5y = ex\]

Super Position

It allows us to decompose a problem into smaller, simpler parts and then combine them to find a solution to the original problem.

Solutions

Explicit

\[F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots, \frac{d^ny}{dx^n}\right) = 0\]

Then the solution of the form \(y = f(x)\) is called explicit solution.

Example

For the following differential equation

\[\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0\]

The solution is

\[y = xex\]

Implicit

A relation \(G(x, y)\) is known as an implicit solution of a differential equation, if it defines one or more explicit solution on \(I\).

Example

For the differential equation

\[y^\prime = - \frac x y\]

the solution is

\[x^2 + y^2 - 4 = 0\]
\[y = \pm \sqrt {4 - x^2}\]

References

Read more about notations and symbols.

  1. Read more about derivatives. ↩↩↩

  2. Read more about partial derivatives. ↩

  3. Read more about functions. ↩

  4. Read more about intervals. ↩↩