ordinary di erential equations. week 1roquesol/math_308_fall_2019...week 1 introduction table of...

Introduction

Ordinary Differential Equations. Week 1

Dr. Marco A Roque Sol

08/27/2019

Dr. Marco A Roque Sol Ordinary Differential Equations. Week 1

Introduction

Table of ContentsFamous QuoteDefinitionsClassificationBasic Examples

Table of contents

1.- Introduction2.- First Order Differential Equations3.- Second Order Differential Equations4.- Higher Order Differential Equations5.- Series6.- The Laplace Transform7.- System of First Order Linear Differential Equations8.- Nonlinear Differential Equations


Introduction


Famous Quotes

Figure: Sophus Lie.


Introduction


Definitions

Differential Equations

A differential equation is any equation which contains derivatives,either ordinary or partial derivatives of an unknown function.

Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it, havingthe form:

F(y (n), y (n−1), ..., y ′, y(t), t

)= 0

Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.


Introduction


Definitions

There many situations where we can find such an object. Thus, forinstance in the case of the study of Classical Mechanics in Physics,if an object of mass m is moving with acceleration a and beingacted on with force F then Newtons Second Law tells us.

F = ma

To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.

a =dv

dtor a =

d2u

dt2

Where v is the velocity of the object and u is the position functionof the object at any time t. We should also remember at this pointthat the force, F may also be a function of time, velocity, and/orposition.


Introduction


Definitions

mdv

dt= F (t, v) or m

d2u

dt2= F (t, u, v)

More examples of differential equations

2y ′′ + 3y ′ − 5y = 0

cos(y)d2y

dx2− (1 + y)

dy

dx+ y3e−y = 0

y (4) + 5y ′′′ − 4y ′′ + y = sin(x)

a2∂2u

∂x2=∂u

∂t

a2∂2u

∂x2=∂2u

∂t2

∂u3

∂2x∂t= 1 +

∂u

∂yDr. Marco A Roque Sol Ordinary Differential Equations. Week 1

Introduction


Definitions

Solution

A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information about thesolution.


Introduction


Classification

OrderThe order of a differential equation is the largest derivative presentin the differential equation. The equation

mdv

dt= F (t, v)

is a first order differential equation, the equations

md2u

dt2= F (t, u, v)

2y ′′ + 3y ′ − 5y = 0

cos(y)d2y

dx2− (1 + y)

dy

dx+ y3e−y = 0


Introduction


Classification

a2∂2u

∂x2=∂u

∂t

a2∂2u

∂x2=∂2u

∂t2

are second order differential equations, the equation

∂u3

∂2x∂t= 1 +

∂u

∂y

is a third order differential equation and finally, the equation

y (4) + 5y ′′′ − 4y ′′ + y = sin(x)

is a fourth order differential equation.

Note that the order does not depend on whether or not you’ve gotordinary or partial derivatives in the differential equation.


Introduction


Classification

Linear Differential Equations

A linear differential equation is any differential equation that canbe written in the following form.

an(t)y (n) + an−1(t)y (n−1) + ...+ a1(t)y ′ + a0(t)y = g(t)

The important thing to note about linear differential equations isthat there are no products of the function and its derivatives, andneither the function or its derivatives occur to any power otherthan the first power.


Introduction


Classification

The coefficients an(t), ..., a0(t), g(t) can be any function. Only thefunction y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only

cos(y)d2y

dx2− (1 + y)

dy

dx+ y3e−y = 0

is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .


Introduction


Basic Examples

1.- Verify that y1(t) = e−3t and y2(t) = et are solutions of theordinary differential equation (ODE) y ′′ + 2y ′ − 3y = 0.

Solution


Introduction


Basic Examples

y2(t) = et


Introduction


Basic Examples

2.- Verify that u(x , t) = e−4tsin(x) is a solutions of the PDE4uxx − ut = 0.

Solution


Introduction


Basic Examples

3.- Determine the values of r for which the differential equationy ′′′ − 3y ′′ + 2y ′ = 0 has a solution of the form y = ert .

Solution


Introduction


Basic Examples

4.- Determine the values of r for which the differential equationt2y ′′ − 4ty ′ + 4y = 0 has a solution of the form y = tr for t > 0.

Solution


Introduction


Basic Examples

5.- Classify each differential equation as an ordinary or partialdifferential equation. Determine the order of each differentialequation and whether the equation is linear or nonlinear.

(1 + y2)d2y

dt2+ y dy

dt + y = et

d3ydt3

+ t dydt + (cos2 t)y = t3

Uxxx − 3uyy + uy + 4u = 0

ut + uuy = 1 + uyy


ordinary di erential equations. week 1roquesol/math_308_fall_2019...week 1 introduction table of...

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