ComE 312 Unit 1

Linear DE of 2nd OrderComE 312 Advanced Engineering Mathematics

Outline

Homogeneous Linear Equations of Second Order

2nd Order Homogeneous Equations with Constant Coefficients

2nd Order Nonhomogeneous Equation by Undetermined Coefficients

Second and Higher Order DE

Linear Differential Equations

Easy to handle

Standard methods of solution

Many practical applications

Nonlinear Differential Equations

Generally more difficult to solve

Homogeneous Linear Equations of Second Order A second-order DE is called linear if it can be written

+ + = ()

and nonlinear if not.

Homogeneous Linear Equations of Second Order If = 0, then the previous equation becomes

+ + = 0

and is called homogeneous. If 0, it is called nonhomogeneous. and are called coefficients of the equation.

Examples: + 4 = sin

+ 2 + 2 = 0

= 2 + 1 (1

Homogeneous Linear Equations of Second Order Superposition or Linearity Principle

= and = are solutions to the homogeneous linear differential equation

= 0

Substituting into the equation we get = therefore, = 0.

Now, let us try = 3 +2

5,

3 +2

5

3 +2

5 = 3 +

2

5 3 +

2

5 = 0

We can obtain new solutions from known solutions by multiplication of constants and by addition.

= 11 + 22

Homogeneous Linear Equations of Second Order Definitions.

General solution. A known function (on an interval I)that satisfies the differential equation. The form is generally, = 11 + 22 where 1& 2 are arbitrary constants.

Particular solution. A solution where the arbitrary constants are assigned a specific value.

Homogeneous Linear Equations of Second Order Initial Value Problem or IVP.For a second order homogeneous linear equation, a general solution will be of the form, = 11 + 22. An initial value problem consists of a DE and two initial conditions

0 = 0 0 = 1

From the given value of 0 we can determine the values of the arbitrary constants.

Example:Solve the IVP,

= 0; 0 = 4, 0 = 2

Outline




Second-Order Homogeneous Equations with Constant Coefficients Let us determine a method for solving homogeneous linear equations

with the form, + + = 0

whose coefficients & are constant. Let us try the function = , as a solution. With the derivatives, = and = 2. Substituting into the DE,

2 + + = 0 2 + + = 0This equation is called the characteristic equation or auxiliary equation. The roots of which are

1 =1

2 + 2 4 , 2 =

1

2 2 4

This gives us the solution to the equation,

1 = 1 and 2 =

2

Second-Order Homogeneous Equations with Constant Coefficients The solution/function are dependent on the sign of the discriminant.

2 4 > 0 2 4 = 0

2 4 < 0

Second-Order Homogeneous Equations with Constant Coefficients Case I. Two distinct real roots

The general solution has the form = 1

1 + 22

General Steps

Step 1. Determine the characteristic equation or the auxiliary equation. Do this by replacing the derivatives with another symbol (r,m or ).

Step 2. Use algebraic techniques to determine the value of (r,m or ). Then substitute into the general solution.

Step 3. If it is an IVP problem, apply the initial conditions to get the particular solution.

Second-Order Homogeneous Equations with Constant Coefficients Case II. Real double root

The general solution has the form = 1 + 2

General Steps




Second-Order Homogeneous Equations with Constant Coefficients Case III. Complex conjugate root

The general solution has the form = 1 cos + 2 sinx

General Steps




Exercise Problems

1. 25 + 40 + 16 = 02. 16 8 + 5 = 03. 92 = 04. 2 2 + 2 = 05. 9 + 6 + = 0; 0 =

3, 0 = 13

36. 25 = 0; 0 = 0, 0 =

207. 2 = 0; 0 =

4, 0 = 17

= 1 + 2 0.8

= 4 1 cos 2 + 2 sin 2

= 1e3 + c2e

3

= 1 + 2 2

= 4 3 3

= 25 25

= 3 72

Outline




Nonhomogeneous Equations with Constant Coefficients A general linear second-order DE has the form,

+ + = ()

If & are constants, the equation takes the form,

+ + = ()

A general solution of this equation is = +

Where,

Method of Undetermined Coefficients

Example:

Solve the IVP, + 2 + 101 = 10.4 , 0 = 1.1, 0 = 0.9

Solution:

The general solution is

= +


Term in () Choice for

( = 0,1, ) + 1

1 ++1 + 0

cos cos + sin

sin

cos cos + sinx

sin


Rules for Method of Undetermined Coefficients

Basic Rule. If r(x) is one of the functions in the first column of the previous table, choose the corresponding function in the second column and determine its undetermined coefficients by substituting and its derivatives into the DE.

Modification Rule. If a term in your choice for happens to be a solution of the homogeneous equation corresponding to the DE, then multiply your choice of by x (or by 2 if the solution corresponds to a double root of the characteristic equation of the homogeneous equation).


Rules for Method of Undetermined Coefficients

Sum Rule. If r(x) is a sum of functions in several lines of the table, first column then choose for the sum of the functions in the corresponding lines of the second column.


Examples:

Solve the nonhomogeneous equations:

+ 4 = 82

3 + 2 =

+ 2 + = , 0 = 1, 0 = 1

+ 2 + 5 = 1.250.5 + 40 cos 4 55 sin 4 , 0 = 0.2, 0 = 60.1


General Steps to the Method:

Step 1. Solve the corresponding homogeneous equation by setting, =0, or find out . You have to rely on the techniques from the previous topic.

Step 2. Solve for , observe the form of () and then look it up on the table for the appropriate . Remember the 3 rules when you choose .After that, differentiate twice

. Substitute into the given DE

to determine the unknown coefficients.

Step 3. If it is an IVP problem, apply the initial conditions to get the particular solution of the given equation.

Exercise Problems

1. + 6 = 63 + 32 +6

2. + 2 + 10 = 252 + 3

3. + 10 + 25 = 5

4. + 1.5 = 122 + 63 4; 0 = 4, 0 = 8

5. 4 = 2 2; 0 =0, 0 = 0

6. + 9 = 6 cos 3 ; 0 =1, 0 = 0

= 13 + 2

2 + 3

= 1 cos 3 + 2 sin 3 +5

22

= 15 + 2

5 +0.525

= 42 + 4

= 1

8sinh 2 +

1

2

1

42

= cos 3 + sin 3

End of Unit 1

ComE 312 Unit 1

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