Chapter 6 - Differential Equations

8/12/2019 Chapter 6 - Differential Equations

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Equation contains an unknown function and one or more of

its derivatives.

Represents a relationship between 2 variables,xandy.

Classification of differential equation:

1. Order : 1storder, 2ndorder,

2. Linearity : Linear or non-linear

3. Homogeneity: Homogeneous or non-homogeneous.

To solve a differential equation is to find all possible

solutions of the equation (y =f(x))

An Initial Value Problem(or IVP) is a differential

equation along with an appropriate number of initial

conditions. For example:


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Show that is the solution to

SolutionWe will first find the first and second derivatives:

Substitute these into the differential equation

So, does satisfy the differential equation and hence

is a solution.


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If is the solution to ,

show that are the initial

conditions to the differential equation. Solution

From previous example, we know that is the

solution to the differential equation. Hence,

So this solution also meets the initial conditions .


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Orderis the order of the highest derivative that occurs in the

equation. For example:

A differential equation is linearif the dependent variable and allits derivative occur linearly in the equation. For example:

Both dy/dxandyare linear, so the differential equation is linear.

The termy3is not linear, so the differential equation is not linear.

22

2

5 26 First order differential equation

2 5 26 11 Second order differential equation

dyx ydx

d y dyy x

dx dx

2 Lineardy

x y xdx

23

2

13 Non-linear

d yy x

x dx


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Determine the order and state the linearity of the following

differential equation

1) A Answer: First order, non-linear

2) Sa Answer: Third order, linear

3) As Answer: Third order, non-linear

4) Sad Answer: First order, linear

5) As Answer: First order, non-linear

6) sa Answer: Second order, linear

3 2

3 22 2 sin

d y d y dyx

dx dx dx

ln 0dy

ydx

4

3

3 2 sin

d y dyx

dx dx

22

dyxy x x

dx

sindy y xdx

2

2 2

d yxy

dx


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Homogeneous differential equationsinvolve only derivatives of

yand terms involvingy, and they're set to 0, as in this equation:

Non-homogeneous differential equationsare the same as

homogeneous differential equations, except they can have terms

involving onlyx(and constants) on the right side, as in thisequation:

In general, we can denote a 2nd order homogeneous equation as

While we can denote a 2nd order non-homogeneous equation as

4 22

4 2 0 Homogeneousd y d yx y

dx dx

4 22

4 2 6 3 Non-homogeneous

d y d yx y x

dx dx

'' ( ) ' ( ) 0y p x y q x y

'' ( ) ' ( ) ( )y p x y q x y g x


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Determine the homogeneity of the following differential

equation

1) A Answer: Non-homogeneous

2) Sa Answer: Homogeneous

3) As Answer: Non-homogeneous

4) Sad Answer: Homogeneous

5) As Answer: Non-homogeneous

6) sa Answer: Non-homogeneous

22

dyxy x x

dx

2

2 2

d yxy

dx

'' 2 ' 2 xy y y e

2

2 2 0

d y dyx xy

dx dx

2 2 3'' 2 ' 2x y x x y x y x

'' 2 ' 2 1y y y


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Differential equations may be formed in practice from a

consideration of the physical problems to which they refer. Mathematically, they can occur when arbitrary constants

are eliminated from a given function.

Example:

Consider whereAand Bare two arbitraryconstants.

sin cosy A x B x

cos sindy

A x B xdx

2

2 sin cos sin cosd y

A x B x A x B xdx

2

2

d yy

dx

2

2

0d y

ydx


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Form a differential equation from the function

Solution

We have

From the given equation,

SubstituteAinto the differential equation:


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Solution

We have

The RHS of the function is identical to the original equation


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Solution

We have

To findA,

To find B, substituteAinto dy/dx,

SubstituteAand Binto the given function,


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Solution

We have


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To solve a differential equation, we have to manipulate the

equation so as to eliminate all the derivatives and leave a

relationship betweenyandx.

There are three method to solve a differential equation: Method 1: By direct integration

Method 2: By separating the variables

Method 3: By substitutingy= vx(homogeneous equations)


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Solve the differential equation

Solution


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We will now consider a method of solution that can often

be applied to first-order equations that are expressible in

the form

The name separable arises from the fact that Equation (1)

can be rewritten in the differential form


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Given

Step 1:

Separate the variables above by rewriting the equation in thedifferential form

Step 2:

Integrate both sides of the equation in Step 1 (the left side with

respect toy and the right side with respect tox):

Step 3:

If H(y) is any antiderivative of h(y) and G(x) is any antiderivative of

g(x), then the equation

will generally define a family of solutions implicitly. In some cases

it may be possible to solve this equation explicitly fory.


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Solve the differential equation .

Solution


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Solution


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Solution


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and then solve the initial-value problem ify(0) = 1.

SolutionFory 0 we can write the differential equation as

Solving the initial-value problem


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Solve the initial-value problem

SolutionWe can write the differential equation as

Solving the initial-value problem


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The simplest first-order equations are those that can be

written in the form

Such equations can often be solved by integration. For

example, if

More generally, a first-order differential equation is called

linearif it is expressible in the form

Some examples:


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Many practical problems in engineering give rise to second

order differential equation of the form

Some examples of second order differential equations:


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Homogeneous Equations

Two continuous functionsfandgare said to be

Linearly dependent - if one is a constant multiple of the other.

Linearly independent if neither is a constant multiple of the

other.

THEOREM

Ify1

andy2

are linearly independent solutions of

y+p(x)y+q(x)y = 0

then its general solution is given by

y(x) =C1y1(x) +C2y2(x)

where C1and C2are arbitrary constants.

xxgxxf )(;)( 2

sin ; 3sinf x x g x x


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In this subject, we restrict our attention to 2nd order

differential linear homogenousdifferential equation with

constant coefficients only.

y+py+qy=0(basic formp and qare constant)ay+by+cy=0

ar2+br+ c=0 (auxiliary characteristic equation)


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Thus, the general solution differential equation depends on

the roots of the auxiliary equation such that:General SolutionRoots of 02 cbrar

042 acb

(r1and r2are real and distinct)

042 acb

(r1 = r2= r )

042 acb

(r1and r

2are complex numbers,

i )

xrxr eCeCxy 21 21

xrxr xeCeCxy21

)sincos( 21 xCxCexy x


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SolutionThe auxiliary equation is

The solution to the differential equation is


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Solve the following initial value problem



Solving (1) and (2),


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Solving (1) and (2),


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From (2), . Substituting C2into (1),

Chapter 6 - Differential Equations

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