differential equation

hamzah asyrani sulaiman at www.asyrani.com

topic

• basic concept• first order differential equation• second order linear differential equation with

constant coefficients• the Laplace transform• Fourier series

what is differential equation?

equation involving an unknown function and its derivatives

example𝑑𝑦𝑑𝑥 =3 𝑥−1

𝑥2 𝑑2 𝑦

𝑑 𝑥2+( 𝑑𝑦𝑑𝑥 )

2

+2 𝑦= ln𝑥

𝑑3 𝑦𝑑 𝑥3

−4 𝑥 𝑑2 𝑦

𝑑 𝑥2=cos2 𝑥

𝜕 𝑦𝜕𝑡 +

𝜕 𝑦𝜕 𝑥=4 𝑡2+2𝑥

there are two types of differential equations (DE)

ordinary differential equation

“if the unknown function depends on only one independent variable”

partial differential equation

“if the unknown function depends on two or more independent variables”

𝑑𝑦𝑑𝑥 =3 𝑥−1

𝑥2 𝑑2 𝑦

𝑑𝑥2+( 𝑑𝑦𝑑𝑥 )

2

+2 𝑦= ln𝑥

𝑑3 𝑦𝑑 𝑥3

−4 𝑥 𝑑2 𝑦

𝑑𝑥2=cos2 𝑥

𝜕 𝑦𝜕𝑡 +

𝜕 𝑦𝜕 𝑥=4 𝑡2+2𝑥

ordinary differential equation

partial differential equation

order

“the order of a DE is the order of the highest derivative in the equation”

order?𝑑𝑦𝑑𝑥=3 𝑥−1

𝑥2 𝑑2 𝑦

𝑑 𝑥2+( 𝑑𝑦𝑑𝑥 )

2

+2 𝑦= ln𝑥

𝑑3 𝑦𝑑 𝑥3

−4 𝑥 𝑑2 𝑦

𝑑 𝑥2=cos2 𝑥

𝜕 𝑦𝜕𝑡 +

𝜕 𝑦𝜕 𝑥=4 𝑡2+2𝑥

1st order

2nd order

1st order

3rd order

notation

“you can represented the order by using a simple notation”

notation

𝑦 ′=𝑑𝑦𝑑𝑥=𝑦 ′ (𝑥 )

𝑦 ′ ′= 𝑑2 𝑦𝑑 𝑥2

=𝑦 ′ ′ (𝑥)

𝑦 𝑛=𝑑𝑛 𝑦𝑑𝑥𝑛=𝑦𝑛 (𝑥)

If the independent variable is

𝑦 ′=𝑑𝑦𝑑𝑡 =𝑦 ′ (𝑡)

𝑦 ′ ′= 𝑑2 𝑦𝑑𝑡 2

=𝑦 ′ ′ (𝑡)

𝑦 𝑛=𝑑𝑛 𝑦𝑑𝑡𝑛

=𝑦𝑛 (𝑡)

If the independent variable is

Linear Differential Equations

• A linear differential equation is any differential equation that can be written in the following form

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

*Important:-no products of the function,

y(t), and its derivatives and neither the function or its derivatives occur to any power other than the first power

linearity

“a crucial classification of differential equations is whether they are linear or nonlinear”

linear𝑑𝑦𝑑𝑥 + 𝑦=sin2𝑥

𝑑2 𝑦𝑑𝑥2

+3 𝑥2 𝑦=0

𝑥3 𝑑3 𝑦

𝑑 𝑥3+3 𝑥 𝑑𝑦𝑑𝑥 −5 𝑦=𝑒𝑥

non linear

( 𝑑𝑦𝑑𝑥 )2

+ 𝑦=𝑠𝑖𝑛𝑥

𝑑2 𝑦𝑑𝑥2

+3 𝑥 𝑦2=0

(1+ 𝑦)𝑑3 𝑦𝑑𝑥3

=sin 𝑦

Got the idea? No?

• We try to look again

why linear?

𝑑𝑦𝑑𝑥 + 𝑦=sin2𝑥

𝑑2 𝑦𝑑𝑥2

+3 𝑥2 𝑦=0

𝑥3 𝑑3 𝑦

𝑑 𝑥3+3 𝑥 𝑑𝑦𝑑𝑥 −5 𝑦=𝑒𝑥

It has a form just like the previous one, we have:-y’(t), y(t)

It has y’’(t), and y(t)

It has y’’’(t), y’(t), and y(t)

why non linear?

( 𝑑𝑦𝑑𝑥 )2

+ 𝑦=𝑠𝑖𝑛𝑥

𝑑2 𝑦𝑑𝑥2

+3 𝑥 𝑦2=0

(1+ 𝑦)𝑑3 𝑦𝑑𝑥3

=sin 𝑦

( 𝑦 ’ (𝑡 ) )2

( 𝑦 (𝑡 ) )2

𝑦 (𝑡 ) 𝑦 ′ ′ ′ (𝑡)

how to detect?

“the dependent variable y and all its derivatives are of the first degree”

“the coefficient depends on constants or independent variables or both of them”

Solution to a differential equation

• A solution to a differential equation on an interval is any function which satisfies the differential equation in question on the interval .

• It is important to note that solutions are often accompanied by intervals and these intervals can impart some important information about solution.

example

• Show that

is a solution to

Solution

• We’ll need the first and second derivatives to do this

Solution

• Plug these as well as the function into the differential equation.

Solution

• So, does satisfy the differential equation and hence is a solution.

• Why then did I include the condition that x > 0?

RECALL BACK

Solution

• To see why recall that

• In this form, it is clear that we’ll need to avoid x=0 at the least as this would give division by zero.

• And then, x > 0 (forever).

𝑦 (𝑥 )=𝑥− 23= 1

√𝑥3

Initial Value Problem

• Initial condition(s) is a condition or set of conditions that help you to solve the differential equation problem

Initial Condition(s) example

• Show that

is a solution to

where

Interval of Validity

• The interval of validity for an Initial Value Problem with initial condition(s)

And/or

Is the largest possible interval on which the solution is valid and contains

We will see it when we actually solve the differential equation (trust me I will tell you)

General Solution of DE

• The general solution to a differential equation is the most general form that the solution can take and doesn’t take any initial conditions into account.

• Just a regular DE

Actual Solution of DE

• What is the actual solution to the following IVP?

Is the general solution to

where

Actual Solution of DE

Put in

So, the actual solution to the IVP is

Implicit/Explicit Solution

Explicit solution“is any solution that is given in the form ”

Implicit solution“any solution that is not in explicit form”

Explicit function

• When an equation can be written in the form , It is said to be an explicit function of x.

• Examples

Implicit function

• Sometimes with equations involving, say , y and, it is impossible to make the subject of the formula. This is called implicit function

• Examples

Homogeneous Function

How to detect homogeneous or not

Given that Then, we put for any power of x and y

Separate it

Arrange it back so is the only remainder

Then, it can be called as homogeneous function as we still got the same equation as

Homogenity

• Homogenity

• HOMOGENEOUS, 2ND ORDER, AND LINEAR

• NON HOMOGENEOUS, 3RD ORDER, AND NON

LINEAR