IZTECH Math255 Notes p1

By E. Z.

Introduction

Dynamic System Undergoes Rate of Change of ...
Heat Diffusion Temperature
Wave Propagation Energy of Wave
Population Growth/Decay Population Size
Object Motion Displacement/Velocity
Definition
\[F(x, y(x), y'(x), y''(x), \ldots, y^{(n)}(x)) = 0\]

Example from Physics

\[\vec{F}_{net}=k\vec{v}(t)\] \[\bigtriangleup\] \[|\] \[\bigcirc\] \[|\] \[\bigtriangledown\] \[\vec{F}_{net}=- m\vec{g}\] \[\]

\[\vec{F}_{net}=m \vec{a}\]

\[k\vec{v}(t) - m\vec{g} = m \vec{a}\]

\[k {\ddot{y}}(t) - m {\dot{y}}(t) - m\vec{g} = 0\]

Classification

By Number of Independent Variables

# of Independent Variables Short Name Full Name
=1 ODE Ordinary Differential Equation
>1 PDE Partial Differential Equation

By Linearity

Name Condition
Linear In the form of: \[ \sum a_{n} \cdot y^{(n)} = g(x) \]
Non-linear Not in the form of: \[ \sum a_{n} \cdot y^{(n)} = g(x) \]

Homogeneous Differential Equations

A DE is said to be homogeneous iff it can be written in the form:

\[\ f(x, y) dy = g(x, y) dx \]


In other words iff expression below can be applied then the DE said to be homogeneous \[\ y'= \frac {y}{x} \]

Solutions of ODE's

Theorem

Let 𝕀 ⊂ ℝ an open interval & consider DE:

\[ \small F(x, y(x), y'(x), y''(x), \ldots, y^{(n)}(x)) = 0\]
Suppose a function ϕ where ϕ ⊂ 𝕀 with properties:
-All derivatives of ϕ up to order n is continous on 𝕀 -ϕ must satisfying the equation
Then 𝕀 is said to be interval of existance

Example

\[\ y''+y=0 \] Show that ϕ = \(\ \small sin(x) \).

Definition
General solution is a way of showing relations between dependent and independent variable that covers ALL possible solutions.

Example: General solution of

\(\ \small y'=2y \),
\(\ \small y=n*e^x \),
\(\ \small n ∈ ℝ \)

Initial Value Problems
(IVP's)

Definition
Initial value problems are given: \[ \small F(x, y(x), y'(x), y''(x), \ldots, y^{(n)}(x)) = 0\] with \[\ \small y(a)= α _0 \] \[\ \small y'(a)= α _1 \] \[\ \small y''(a)= α _2 \] \[\ \small y^{(n)}(a)= α _{n-1} \]

First Order ODE's

Definition
\[\ f(x, y, y')=0 \]

Seperable Equations

Definition

Functions that can be written in the form:

\(\ \small \frac{dy}{dx}=f(x) . g(y) \) are said to be seperable DE's.

Equations that can be transfromed into seperable DE's

Definition
If a function is not in the form of:
\(\ \frac{dy}{dx}= f(x). g(y) \)
but can be transformed into \(\ \frac{du}{dx}= f(x). g(u) \) by using transformation \(\ u(x).x= y(x),\) \(\ y'=u' .x + u \)

Linear First Order DE's

Definition
\[\ y'+p(x)y=q(x) \]
By using method of multiplying both sides with μ(𝓍)=ℯ ∫ p(𝓍) d𝓍 ,
the DE become (μ(𝓍).𝓎)'=𝓆(𝓍)