PCL0016 CALCULUS

ONLINE NOTES

Topic 6

Differential Equations

Faculty of Engineering,Multimedia University,

Persiaran Multimedia, 63100 Cyberjaya,Selangor, Malaysia.

Faculty of Engineering(FOE)

PCL0016 Topic 6

CHAPTER 6: DIFFERENTIAL EQUATIONS

Objectives: To be able to understand the concept and solve the problems of the followings

1. First-Order Differential Equationsa) Separable Equationsb) Linear Equations

2. Second-Order Differential Equationsa) Homogenous Linear Equationsb) Non-homogenous Linear Equations

Contents:6.1 Introduction to Differential Equations6.2 Formation of Differential Equations6.3 Solution of Differential Equations6.4 First Order Differential Equations6.5 Second Order Differential Equations

6.1 INTRODUCTION TO DIFFERENTIAL EQUATIONSA differential equation is an equation that contains an unknown function and its

derivatives - an equation in which at least one term contains , , etc.

Classification of differential equation:1. Order: First order, second order, …2. Linearity : Linear or non-linear3. 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:

Example 1:

Show that is the solution to

Solution:We will first find the first and second derivatives:

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PCL0016 Topic 6 Substitute these into the differential equation

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

Order is the order of the highest derivative that occurs in the equation. For example:

1. First order differential equation

- it contains only first-order derivatives, - it is called a linear differential equation

2. Second order differential equation

- it contains a second order derivatives - this is not a linear differential equation

A differential equation is linear if the dependent variable and its entire derivative occur linearly in the equation. For example:

Both dy/dx and y are linear, so the differential equation is linear.

The term y3 is not linear, so the differential equation is not linear.

Homogeneous differential equations involve only derivatives of y and terms involving y, and they're set to 0, as in this equation:

Non-homogeneous differential equations are the same as homogeneous differential equations, except they can have terms involving only x (and constants) on the right side, as in this equation:

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

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

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PCL0016 Topic 6

6.2 FORMATION OF DIFFERENTIAL EQUATIONS

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 where A and B are two arbitrary constants.

Example 2:

Form a differential equation from the function

Solution:

We have

From the given equation,

Substitute A into the differential equation:

Example 3:Form a differential equation from the function

Solution:

We have

The RHS of the function is identical to the original equation

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Example 4:Form a differential equation from the function

Solution:

We have

To find A,

To find B, substitute A into dy/dx,

Substitute A and B into the given function,

6.3 SOLUTION OF DIFFERENTIAL EQUATIONS

A function f is called a solution of a differential equation if the equation is satisfied when and its derivatives are substituted into the equation. To solve a differential

equation, we have to manipulate the equation so as to eliminate all the derivatives and leave a relationship between y and x.

There are three methods to solve a differential equation:1. Method 1: By direct integration2. Method 2: By separating the variables

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PCL0016 Topic 6 3. Method 3: By substituting y = vx (homogeneous equations)

A) Method 1 : By Direct Integration

A differential equation is expressed

To solve the differential equation, we can use direct integration:

B) Method 2 : By Separating the Variables

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

The method 2 is used when the equation is in the simplest first-order form of equation,

the expression for can be factored as a function of x times a function of y.

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To solve the equation:1. Separate the variables y from x, i.e., by collecting on one side all terms involving y

together with dy, while all terms involving x together with dx are put on the other side.

2. Integrate both sides of the equation.3. If the solution can be defined explicitly, i.e., it can be solved for y as a function of x,

then do it. If not, the solution can be defined implicitly, i.e., it cannot be solved for y as a function of x.

Examples 5:Solve the following differential equations:

Note:1. Solutions for the above examples are called the general solutions.

2. The general solution for example 1 is defined explicitly, where as, the general solution for example 2 is defined implicitly.

3. If C is to be evaluated, then the solution is called the solution of the initial-value problem.

C) Method 3 : By Substituting y = vx (Homogeneous Equations)

Let’s consider the following equation:

This looks simple but we find that we cannot express the RHS in the form of “x-factor” and “y-factor”, so we cannot solve by method of separating the variables.In this case, we make substitution y = vx, where v is a function of x. Differentiate with respect to x, we have

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PCL0016 Topic 6 Substitute into the differential equation, we have

Rearranging the equation produces

The above equation is now expressed in terms of v and x. Now we can use the method of separating the variables:

Example 6:Solve the differential equation

Solution:

We have ------- (1)

Rearranging equation yields

Substituting (1) into above equation

Separating variables

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Example 7:

Solve the differential equation .

(Hint: )

Solution:

We have ------- (1)

Rearranging equation yields

Substituting (1) into above equation

Separating variables

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PCL0016 Topic 6 6.4 FIRST-ORDER DIFFERENTIAL EQUATIONS

This method is used when the equation is in the form of linear equation in which the variables cannot be separated.

(Basic form)

where and – continuous functions on a given interval

Examples:

1.

2.

3.

Procedures to solve a linear differential equation:

1. Must remember the basic form :

2. Multiply both sides by the integrating factor gives :

where

3. Integrate both sides of the equation obtained in (2) and the result obtained is :

4. Finally, solve for y. Be sure to include the constant of integration in this step.

Example 8:

Solve the following differential equations:

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6.5 SECOND-ORDER DIFFERENTIAL EQUATIONS

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PCL0016 Topic 6 Second-order linear differential equation has the following basic equation:

or in alternative notation,

where P, Q, R and G are continuous functions.

If for all x, then, the equation

is said to be homogeneous.

If for all x, then, the equation

is said to be nonhomogeneous.

Examples of second-order homogeneous differential equations:

Examples of second-order nonhomogeneous differential equations:

Solving Second-Order Linear Homogeneous Differential Equations With Constant Coefficients.Two continuous functions f and g are said to be linearly dependent if one is a constant multiple of the other. If neither is a constant multiple of the other, then they are called linearly independent.

Examples: and => linearly dependentand => linearly independent

The following theorem is central to the study of second-order linear differential equations:

Theorem:

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PCL0016 Topic 6 If and are linearly independent solutions of the homogeneous equation

(1)

then the linear combination (2)is the general solution of equation (1) where and are arbitrary constants.[Every solution of equation (1) can be obtained from (2) by choosing appropriate values for the arbitrary constants and ].

We restrict our attention to second-order linear homogeneous differential equations with constant coefficients (the coefficient functions P, Q, and R are constant functions).

or

where a, b, and c are constants, .

Replacing with , with , and with , we will obtain the auxiliary equation:

Thus, the general solution of the second– order linear differential equation depends on the roots of the auxiliary equation .

Case I: (the roots and are real and distinct)then

Case II: (the roots are real and equal one real root r)then

Case III: (the roots are complex number, and )then

Note:Each of these solutions contains two arbitrary constants and since the solution of a second–order differential equation involves 2 integration.

Example 9:

Solve the following differential equations:

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