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1

What is a differential equation?

du/dt = cos(t)

d2u/dt2 = u( du/ft) + tu^2

∂u/∂t = ∂2u/∂x2 - ku

Independent variables?

What is a differential equation? A relationship between a function and it's derivatives

Ordinary differential equations

du/dt = cos(t)
First order linear inhomogenous ode
d2u/dt2 = u( du/dt) + tu^2
Second order non linear homogeneous

Partial differential equation
∂u/∂t = ∂2u/∂x2 - ku
Second order linear homogeneous pde

Partial differential equations are used with more than one independent variable. Here x and t are independent variables

2

creating an ode given data/ question rates

The speed of/ the rate of is the change over time so = du/dt

Negative positive if decreasing increasing

du/dt is a function of the dependent variable

Takes a particular value when t is specified
Eg in the reals

3

Identifying dependent and independent variables

What are they?

The dependent variable- u,y
The variable being differentiated by the independent variable e.g. x, t

The independent variable-
Ode has one independent variable
Pde is of two or more

4

The state of the system

The state of the system is the collection of all the relevant dependent variables for a system

Eg variable variables

Eg state would be number of individuals in each population. Temperature and pressure if a gas

5

A model of a system

A model of the system is a mathematical representation of the system that allows the dynamics evolution of the system state to be calculate

6

Example- deriving the ode equation

Consider a population. Let the number at time t be represented by continuous variable n(t). Suppose the organisms reproduce by division at rate bn(t). Then we model by •1)

•2) if organisms die of natural causes at rate d_0 n(t)

•3) if competition due to resources causes death at a rate of d_c [n(t)]^2

•1) dn/dt = bn(t)
Rate of change of population size = Birth rate times size of the population

•2) dn/dt = bn(t) - d_0 n(t)
Rate of change of population size = birth rate times size of pop- death rate times size of pop
•3) dn/dt = bn(t) - d_0 n(t) - d_c [n(t)]^2
Rate of change of population size = birth rate times size of pop- death rate times size of pop - death rate from competition times size of population squared

Notation: dn/dt = b_n _d_0*n -d_c*n^2
n(t) continuous variable with independent variable t

7

Example constructing Newton's second law of motion
Ode

Position of a point of mass m constrained to move in one dimension x under the action of a force F(x,t) is governed by Newton's second law of motion: ma =F(x,t) where a is the acceleration of the particle. Recall that acceleration is just the change in velocity over time so a=dv/dt

where v is the velocity. But velocity is the change in distance over time..

v=dX/dt

a=dv/ft = d/dt (v) = d2x /dt2

Combining

m( d2x/dt2) = F(x,t)

8

Separation of variables

Dividing by dependent variable
Don't forget constants

Eg n is always positive as represents number so can divide by


dn/dt =nt

Seperation solving gives general solution n=Aexp(bt)
Exponential growth
Determine a by ínitial condition

Eg exponential decay du/dt= -ku(t)
u(t) =u_0 exp(-kt)
Time for half to decay is ln 2/k

9

Deterministic models

Deterministic models where once we have specified an initial value for the dependent variable the differential equation allows us to

Determine it's value for all subsequent times


Eg exponential growth and decay after specified initial value for u

10

Classification of ODEs

Order
Linearity
Homogeneity

Order of a differential equation is the order of the highest derivative of the dependent variable that appears in the equation

Eg not (du/dt) ^2 but d2u/dt^2

Linearity: a differential equation of the dependent variable u is said to be linear if it satisfies the following two conditions:
1) the only u-dependent terms are u Itself and derivatives of u
2) u and it's derivatives do not appear multiplied together

Eg sinx du/dx =u is linear
d2u/dt2 + x^2( du/dt)^2 + exp(-x) =0
Non linear (2nd term)
Remember only related to the dependent variable and not the independent

Homogeneity: remember might have to reorder
The general nth order linear differential equation can be written as d^n u /dx^n + a_n-1 (x) d^(n-1) u /dx^(n-1) + ... + a_2(x) d^2u /dx^2 + a_1(x) du/dx + a_0(x)u =b(x)
Homogeneous if 0 Inhomogeneous if not

Ie function of independent variable on its own means inhomogeneous

11

First order odes

dy/dX = f(x)

y(x) = integral( f(x).dx +c )

Or


dy/dx = f(x,y)
Since the the integration of the right hand side would require the knowledge of y(x), which is the solution
We use different methods

12

Solving
dy/dx = f(x,y)

First order ode

•if f(x,y) = -p(x)y + q(x) for functions then the equation is a linear first order ode
dy/dx + p(x)y = q(x)

**** if homogeneous then seperation of variables gives y(x) = exp( - int( p(x).dx +c)
= A exp(-integral(p(x).dx)
Where A= exp(c)

Check constants out

**** if inhomogeneous then use product rule and integrating factor P(x) = exp( integral( -p(x).dx))

•first one:the equation du/ex =f(x,u) might be seperable

13

Seperable first order ODE

A differential equation such as
du/dx = f(x,u) is termed separable if f(x,u) is a product of functions of x and u i.e. f(x,u) = g(x) h(u) thus seperating and integrating gives...


H(u) = integral( g(x).dx +c) where c is a constant of integration and H= integral( 1/h(u) .du)

If possible to invert H(u) we can rearrange the equation to find a solution u in terms of x

Just keep track of independent and dependent variables

14

Long term behaviour:

If certain constants positive negative? Question in practise