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Chapter 2 Second order differential equations

Page history last edited by sidjaggi 13 years, 5 months ago

Chapter 2 Second order differential equations

 

1 Basic concepts

 

1.1 Linear ODEs

 

A second-order linear ODE is of the form

 

                                                      Formula                                                      (1)

 

Note: If the equation has, say Formula as the first term, then divide by Formula to get to the standard linear form (1) with Formula as the first term.
For example, Formula has a standard linear form Formula.

 

Question: What would the standard linear form for a third order differential equation be? How about a higher order differential equation?

 

1.2 Linear homogeneous ODEs

 

If Formula,

 

                                                      Formula                                                     (2)

 

Example 1.2.

 

(a) Formula is a nonhomgeneous linear ODE.
(b) Formula is a homogeneous linear ODE. The standard form is Formula.
(c) Formula is a nonlinear ODE.

 

2 Homogeneous Linear ODEs


2.1 Superposition Principle


If Formula, Formula are two solutions of a linear homogeneous ODE, then any linear combination of them is too.
That is,


                                                     Formula


is also a solution of that ODE.

 

Example 2.1.

 

The functions Formula and Formula are solutions of the homogeneous linear ODE Formula for all Formula. We can verify this by differentiation and substitution. We multiply Formula by any constant, for instance, Formula, and Formula by, say, Formula, and take the sum of the results, claiming that it is a solution. Indeed, differentiation and substitution gives

 

                                                     Formula

 

Note: By careful that the superposition theorem holds for homogeneous linear ODEs only but does not hold for nonhomogeneous linear or nonlinear ODEs.

 

Exercise: PROVE that any linear combination of solutions of a linear homogenous ODE is also a solution. Does your proof work for a non-homogenous ODE? Or for a non-linear ODE?

 

 

2.2 Linear independence


Two functions Formula and Formula are said to be linearly independent on an interval Formula where they are defined if

 

                                                     Formula

Then if Formula or Formula, we can divide and see that Formula and Formulaare proportional,

 

                                                     Formula 

 

 

In contrast,  Formula and Formula are said to be linearly dependent on Formula if the above equation also holds for some constants Formula and Formula not both zero. 

Thus, in the case of linear independence the two functions are not proportional.

 

Example 2.2.


(a) Formula and Formula are linearly independent since Formula only true when Formula and Formula are both zero. Or we can say their quotient is Formula(or Formula).
(b) Formula and Formula are linearly dependent since Formula can be true when Formula and Formula.

 

Exercise 2.2.

 

Are the following functions linearly independent on the given interval?
(a) Formula
(b) Formula
(More Exercises: Problem Set 2 question 1)

 

 

 

2.3 General solution/Basis/Particular solution

 

A general solution Formula of (2) has the form


                                                     Formula

 

  • The function Formula is on an open interval Formula.

 

  • The functions Formula and Formula are linearly independent on Formula.

 

  • The constants Formula and Formula are arbitrary.

 

  • The functions Formula, Formula are called a basis of solutions of (2) on Formula.

 

  • Choosing Formula, Formula gives particular solutions.


2.4 Initial value problem


An initial value problem consists of (2), along with two initial conditions (since this is a second order ODE), which may be of the form


                                                     Formula

 

  • The given values Formula and Formula are at the same given Formula on Formula
  • These are used to determine Formula and Formula in a general solution


Example 2.4.

 

Consider Formula and Formula,

Assume we know that Formula and Formula are solutions of this equation, the general solution for this is given by:


                                                     Formula


From the 1st initial condition of Formula, Formula; From the 2nd initial condition of Formula, Formula.
Therefore, we have Formula, Formula, and we have a particular solution Formula.

 

Exercise: What if, instead of knowing y(0) and y'(0), we knew y(0)=4 and y(1)=e+3/e -- how would you find the particular solution? How about if we knew y(0)=4 and y'(1) = e-3/e?

 

 

3 Homogeneous linear ODEs with constant coefficients


Consider second-order homogeneous linear ODEs whose coefficients Formula and Formula are constant,


                                                     Formula                                                     (6)


3.1 Solving homogeneous linear ODEs with constant coefficients


Guess the solution to be


                                                     Formula

 

then substitute it into (6), we get


                                                     Formula

 

Hence if Formula is a solution of the characteristic equation


                                                     Formula                                                    (7)


From algebra we further know that the quadratic equation (7) may have three kinds of root, depending on the sign of the discriminant


                                                     Formula

 

This leads to table below,

 

Case
Root of (7)
Basis of (6)
General Solution of (6)
Formula  Distinct real FormulaFormula
Formula, Formula
Formula 
Formula  Real double root Formula
FormulaFormula
Formula 
Formula  Complex conjugate  Formula, Formula
Formula, Formula
Formula 

 

Note: For the case  Formula, we get only one root. To obtain a second independent solution, we use the method of reduction of order (see advanced topics at the bottom of this worksheet).
Note: For the case of Formula, the roots of (7) and thus the solutions of (6), at first seem to be complex-valued. However, we can obtain a basis of real solutions Formula, Formula where Formula.

Note: Euler's formula Formula


Example 3.1.

 

Consider the ODE


                                                     Formula

 

(a) When Formula (in this case we choose Formula), Formula, the basis will be Formula, Formula, the general solution is Formula.
(b) When Formula, Formula, the basis will be Formula, Formula, the general solution is Formula.
(c) When Formula (in this case we choose Formula, Formula, the basis will be Formula, Formula, the general solution is Formula.

 

Exercise 3.1.

 

(a) Verify the solution of Example 3.1 (b) and (c).
(b) Solve the ODE Formula

(More Exercises: Problem Set 2 Question 3(a) 21, 22, 23)

 

 

5 Modeling: Free Oscillations (Mass-Spring System)


This modeling problem is about Simple Harmonic Motion (SHM). Its model is a homogeneous linear ODE.
We measure the displacement Formula of the body from the 'equilibrium point', when the system is in static equilibrium (at the origin Formula), with the downward direction regarded as the positive direction, and upward negative.


5.1 Undamped System


If the effect of damping on the system is negligible, we have the equation


                                                     Formula                                                  (10)


where Formula is the mass of the body and Formula is the spring constant.
By the method we discussed above, we obtain the general solution


                                                     Formula                                                  (11)


Since it has the period Formula, the frequency of the oscillation is Formula, which is also called the natural freqency of the system.
The general solution can be changed to


                                                     Formula                                                  (12)


by using Formula . This is called a phase-shifted cosine with amplitude Formula and phase angle Formula.

 

Figure 1: Harmoic oscillations

 

Figure 1 shows typical forms of the general solution, all corresponding to some positive initial displacement Formula and different initial velocity Formula.

 

Example 5.1.

 

Suppose an iron ball has mass Formula, and a spring has a spring constant Formula. Find the natural frequency when the mass-spring system execute. What will its motion be if we pull down the weight an additional Formula and let it start with zero initial velocity?
Solution. The natural frequency Formula. From (12) and Formula and Formula, the motion is


                                                     Formula


5.2 Damping System


When the effect of damping is not negligible, we need to add a damping force and the equation becomes


                                                     Formula                                                 (13)


where Formula is called the damping constant and is always positive (What would the physical significance of it being negative be?)
Consider the characteristic equation of (13),


                                                     Formula                                                 (14)


Then we obtain the roots


                                                     Formula                                                 (15)


As before, there are three cases for the discriminant Formula. Those cases correspond to three types of motion, depending on the amount of damping (high, medium, or low).

 

Case I.
Formula  Distinct real roots Formula, Formula
(Overdamping)
Case II.
Formula  A real double root
(Critical damping)
Case III.
Formula  Complex conjugate roots
(Underdamping)

 

Case
Solution
I
Formula 
II
Formula 
III
Formula 

 

 

Note: About Case III, Formula, Formula and Formula.

 

Figure 2: Typical motion in Case I overdamped case

   
Figure 3: Case II Critical damping
5Figure 4: Damped oscillation in Case III

 

Example 5.2.

 

How does the motion in Example 5.1 change if we change the damping constant Formula to one of the following three values, with Formula and Formula as before?


                                                     Formula


Solution.
Consider the ODE


                                                     Formula

 

For Case (I), Formula, the ODE becomes Formula. The discriminant of the characteristic equation is greater than zero. So, it has two roots -9 and -1. The general solution is

 

                                                     Formula

 

By consider the initial conditions, the solution is

 

                                                     Formula

 

Exercise 5.1.

 

Solve Cases (II) and (III) of the example above.

 

6 Linear Dependence and Independence of Solutions - Wronskian


Let the ODE Formula have continuous coefficients Formula and Formula on an open interval Formula. Then two solutions Formula and Formula of this equation on Formula are linearly dependent on Formula if and only if their "Wronskian"


                                                     Formula

 

is 0 at some Formula in Formula. Furthermore, if Formula at an Formula in Formula, then Formula on Formula; hence if there is an Formula in Formula at which Formula is not 0, then Formula, Formula are linearly independent on Formula.

Note: The claim is true for that Formula and Formula are analytic functions (infinitely differentiable).

 

Example 6.1.

 

A general solution of Formula on any interval is Formula.
Consider Formula and Formula, the Wronskian

                                                     Formula


Exercise 6.1.

 

Show linear independence by Wronskian: Formula, Formula

(More Exercises: Problem Set 2 Question 4)

 

7 Nonhomogeneous ODEs


The nonhomogeneous linear ODE


                                                     Formula                                                (16)


where Formula.


7.1 General solution and particular solution

 

A general solution of the nonhomogeneous ODE (16) on the open interval Formula is a solution of the form

 

                                                     Formula                                               (17)

 

 

  • The function Formula is a general solution of the homogeneous ODE Formula on Formula.

 

  • The function Formula is any solution of (16) containing no arbitrary constants.
  • Choosing Formula, Formula gives particular solutions.

 

8 Solving the nonhomogeneous ODE

 

To solve the nonhomogemeous ODE (16) or an initial value problem for (16), we need to perform the steps below.
Step 1: Find the general solution Formula for the corresponding homogeneous ODE (r(x)=0).
Step 2: Find Formula, as described below. Then we have a general solution Formula (containing arbitrary constants Formula and Formula).
Step 3: Finally, use the initial conditions to find Formula, Formula, which gives the particular solution.

 

 

To find Formula, we use the below method:


8.1 Method of Undetermined Coefficients


The method of undetermined coefficients is suitable for linear ODEs with constant coefficients, say Formula and Formula.


                                                     Formula                                              (18)

 

  • Guess a form of Formula that is ``similar to" Formula, but with unknown coefficients.
  • For example if Formula, guess Formula.
  • Determine the coefficients by substituting that Formula and its derivatives into the ODE


8.1.1 Choice Rules for the Method of Undetermined Coefficients


(a) Basic Rule: If Formula in (18) is one of the functions in the first column in Table, choose Formula in the same line and determine its undetermined coefficients by substituting Formula and its derivatives into (18). (See Example 8.1.1)
(b) Modification Rule: If a term in your choice for Formula happens to be a solution of the homogeneous ODE corresponding to (18), multiply your choice of Formula by Formula (or by Formula if this solution corresponds to a double root of the characteristic equation of the homogeneous ODE). (See Example 8.1.2)

(c) Sum Rule: If Formula is a sum of functions in the first column of the Table below, choose for Formula the sum of the functions in the corresponding lines of the second column. (See Example 8.1.3)

 

Table

Term in Formula
Choice for Formula
Formula  Formula 
Formula  Formula 
Formula  Formula 
Formula 
Formula   Formula
Formula 

 

Example 8.1.1.

 

Application of the basic rule (a) to solve the initial value problem


                                                     Formula

 

Solution.
Step 1. Find the general solution of the homogeneous ODE Formula. We get


                                                     Formula


Step 2. Finding the particular solution Formula.
We first try Formula, Formula. By substitution, Formula. Comparing the coefficient of each power of Formula (Formula and Formula), we get Formula and Formula.

 

But this is a contradiction!

 

The second line in the Table suggests instead the choice


                                                     Formula

 

Comparing the coefficients of Formula, Formula, Formula on both sides, we have Formula, Formula, Formula.
Hence Formula. This gives Formula, and

 

                                                     Formula

 

Step 3: Solution of the initial value problem.
By using the two initial conditions, we get the answer


                                                     Formula

 

Example 8.1.2.

 

Application of the Modification Rule (b)
Solve


                                                     Formula

 

Solution.
Step 1. Find Formula.
Consider the homogeneous ODE Formula,


Ex 8.1.1.

 

First, we find the general solution of Formula.
As before, this is


                                                     Formula


Step 2. Finding Formula.
First consider Formula. But this function is a solution of the homogeneous ODE. According to the Modification Rule, we choose


                                                     Formula

 

We substitute these expressions into the given ODE and factor out Formula. This yields


                                                     Formula

 

Comparing the coefficients of Formula, Formula, and 1, we get Formula. This gives the solution Formula. Hence the given ODE has the general solution

 

                                                     Formula

 

Example 8.1.3.

 

Application of the Sum Rule (c)
Solve


                                                     Formula

 

Solution.
Step 1. Find the general solution of the corresponding homogeneous ODE. (Try it yourself !!!) The general solution should be


                                                     Formula


Step 2. Find Formula.
We write Formula, where Formula corresponds to the exponential term and Formula to the sum of the sum of the other two terms, i.e., the trigonometric function.
We set Formula.

 

Ex 8.1.2.

 

Try to find Formula in this case.
By substitution, we have Formula.
We now set Formula, as in the Table,


Ex 8.1.3.

 

Again we try to find Formula.
Hence Formula. Together,


                                                     Formula

 

Exercise 8.1.

 

Find a general solution of Formula.

(More Exercises: Problem Set 2 Question 5)

 

9 Modeling: Free Oscillations. Resonance

Recall the spring-mass system model in Section 5. We now extend it by including an external force, say Formula, on the right. Then we have


                                                     Formula                                               (20)


The resulting motion is called a forced motion with forcing function Formula, which is also known as the input or driving force to the system, and the solution Formula to be obtained is called the output or the response of the system to the driving force.


Of special interest are periodic external forces, and we shall consider a driving force of the form


                                                     Formula


Then (20) becomes


                                                     Formula                                               (21)


By using the method of undetermined coefficients,


                                                     Formula                                               (22)


By substitution, we get


                                                     Formula

 

Since Formula, then Formula and we obtain


                                                     Formula

 

We thus obtain the general solution of (21)


                                                     Formula                                               (23)


9.1 Undamped forced oscillations. Resonance


In this case, Formula. Since Formula, Formula become


                                                     Formula                                               (24)


Finally, the solution becomes


                                                     Formula                                               (25)


We see that it is a superposition of two harmonic oscillations of the natural frequency.

Resonance: In (24) the maximum amplitude of Formula is


                                                     Formula                                               (26)


The term Formula is called the resonance factor. If Formula, then Formula and Formula tend to infinity. If Formula, resonance occurs. From (26) we see that Formula is the ratio of the amplitudes of the particular solution Formula and of the input Formula. In case of resonance (21) becomes


                                                     Formula                                               (27)

 

Then (24) is no longer valid, and from the Modification Rule we find the particular solution is


                                                     Formula                                               (28)


By substituting this into (27) we get


                                                     Formula                                               (29)


Because of the factor Formula, the amplitude of the vibration become larger and larger. Practically speaking, systems with very little damping may undergo large vibrations that can destroy the system.

 

Figure 5: Particular solution in resonance


Beats: Another interesting type of oscillation is obtained if Formula is close to Formula. Take, for example, the particular solution


                                                     Formula                                               (30)


By using Formula, we can write this as


                                                     Formula                                               (31)


Since Formula is close to Formula, the difference Formula is small. Hence the period of the last sine function is large, and we obtain an oscillation of the type shown in Figure 6, the dashed curve resulting from the first sine factor.

 

Figure 6: Beats

 

9.2 Damped forced oscillations


In this case, Formula -- the damping term cannot be negligible.
After a sufficiently long time the output of a damped vibrating system under a purely sinusoidal driving force will practically be a harmonic oscillation whose frequency is that of the input. Then Formula approaches zero. Hence the "transient solution" Formula approaches the "steady-state solution" Formula.
In this case, the amplitude may have a maximum for some Formula depending on the damping constant Formula. This is called practical resonance.
To study the amplitude of Formula as a function of Formula,


                                                     Formula                                               (32)


Formula is called the amplitude of Formula and Formula is the phase angle or phase log. these quantities are


                                                     Formula                                               (33)


To find the maximum of Formula, we set the derivative of Formula equal to zero and solve for it to obtain


                                                     Formula                                               (34)

 

By reshuffling terms we have


                                                     Formula

 

The right side of this equation becomes negative if Formula, so that then (34) has no real solution and Formula decreases monotonically as Formula increases. If Formula is smaller, Formula, then (34) has a real solution Formula, where


                                                     Formula                                               (35)


Substituting (36) into (33) gives


                                                     Formula                                               (36)


We see that Formula is always finite when Formula. Furthermore, since the expression


                                                     Formula

 

in the denominator of (36) decreases monotonically to zero as Formula goes to zero, the maximum amplitude (36) increases monotonically to infinity, in agreement with result in section 9.1.

 

   
Figure 7: Amplication Formula (ratio of the amplitudes of output and input) as a function of Formula for Formula, Formula, hence Formula, and various values of the damping constant Formula
Figure 8: Phase angle Formula, which is less than Formula when Formula, and greater than Formula for Formula

 

Example 9.1.

 

Consider the mass-spring system with Formula, Formula, and Formula. The general solution to the ODE without external force is


                                                     Formula

 

When we add the external force Formula to start the oscillation, the steady-state solution is


                                                     Formula


Exercise 9.1.

 

Find the transient motion of the mass-spring system modeled by the given ODE


                                                     Formula

 

 

 

 

ADVANCED TOPICS

 

2.5 Finding a basis if one solution is known


We can find the second linearly independent solution by the method of reduction of order. Let Formula be a known solution of (2) on Formula. Guess the second linearly independent solution Formula is of the form Formula,

 

                                                     Formula                                                     (3)


Ex 2.5.1.

 

Prove (3).

Substitute FormulaFormula and Formula into (2), we have


                                                     Formula                                                    (4)


Ex 2.5.2.

 

Prove (4).
Substituting Formula and Formula, (4) becomes


                                                     Formula


Separation of variables and integration gives

 

                                                     Formula                                                   (5)


Ex 2.5.3.

 

Prove (5).
Since Formula, we have Formula, thus


                                                     Formula

 

Example 2.5.

 

Consider the differential equation Formula. One solution of it is Formula. We divide the ODE by Formula to get


                                                     Formula

 

Then we need to find Formula. Because of (5),


                                                     Formula

 

                                                     Formula

 

By partial fractions,


                                                     Formula

Therefore,


                                                     Formula

 

It is obvious that Formula and Formula are linearly independent.


Exercise 2.5.

 

Find Formula of Formula when Formula.

(More Exercises: Problem Set 2 Question 2(a))

 

 

 

4 Euler-Cauchy equations


Euler-Cauchy equations are ODEs of the form


                                                     Formula                                                   (8)


4.1 Solving Euler-Cauchy equations


To find the basis of (8), we substitute


                                                     Formula


and its derivatives into (8). Then we have the characteristic equation


                                                     Formula                                                  (9)


Similar to the case of linear ODEs with constant coefficients, there are three different cases.

 

Case 
Root of (9) 
Basis of (8) 
General Solution of (8) 
Formula  Distinct real Formula,  Formula
FormulaFormula
Formula 
Formula  Real double root Formula
FormulaFormula
Formula 
Formula  Complex conjugate  FormulaFormula
FormulaFormula
Formula 

 

Note: For the case of Formula, the case of complex roots is of minor practical importance.


Example 4.1.


(a) Consider the ODE Formula. The general solution is Formula (by the table above, first row).
(b) Consider the other ODE Formula. The general solution for all positive Formula is Formula since Formula (by the table above, second row).

 

Exercise 4.1.

 

Find a general solution of Formula

(More Exercises: Problem Set 2 Question 3(c))

 

 

8.2 Solution by Variation of Parameters


We can get the particular solution Formula on Formula in the form


                                                     Formula                                               (19)


where FormulaFormula form a basis of solutions of the corresponding homogeneous ODE


                                                     Formula

 

on Formula, and Formula is the Wronskian of FormulaFormula,


                                                     Formula

 

Note: BE CAREFUL: The ODE should be written in the standard form, i.e., if the ODE starts with Formula as in Formula, it should be divided first by Formula to get Formula. Then FormulaFormula and Formula.

 

 


Note: Even though this method seems to be very general and always give the answer via a formula, in practice, the integrations above may often cause difficulties, and so may the computation of FormulaFormula if the ODE has non-constant coefficients. Use the previous methods if they work and you have a choice -- they are simpler!

 

Example 8.2.

 

Solve the nonhomogeneous ODE


                                                     Formula

 

Solution. First, written in standard form, the ODE become


                                                     Formula

 

A basis of solutions of the homogeneous ODE an any interval is FormulaFormula (Example 2.1). This gives the Wronskian

 

                                                     Formula

 

From (19), choosing zero constants of integration,


                                                     Formula

 

                                                     Formula


From Formula and the general solution Formula of the homogeneous ODE we obtain the answer


                                                     Formula

 

Exercise 8.2.

 

Solve Formula.

(More Exercises: Problem Set 2 Question 6)

 

 

Case II
Figure 2: Typicalmotion in
Case I overdamped case
Example 5.2. How does the motion in Example 5.1 change if we change the damping constant c to one of
the following three values, with y(0) = 0:16 and y0
(0) = 0 as before?
(I) c = 100 kg=sec; (II) c = 60 kg=sec; (III) c = 10 kg=sec:
Solution.
Consider the ODE
10y00
+ cy0
+ 90y = 0:
For Case (I), c = 100, the ODE become 10y00
+ 100y0
+ 90y = 0. The discriminant of the characteristic
equation is greater than zero. So, it has two roots -9 and -1. The general solution is
y = c1e 9t
+ c2e t
:
By consider the initial conditions, the solution is
y =  0:02e 9t
+ 0:18e t
:Example 5.2. How does the motion in Example 5.1 change if we change the damping constant c to one of
the following three values, with y(0) = 0:16 and y0
(0) = 0 as before?
(I) c = 100 kg=sec; (II) c = 60 kg=sec; (III) c = 10 kg=sec:
Solution.
Consider the ODE
10y00
+ cy0
+ 90y = 0:
For Case (I), c = 100, the ODE become 10y00
+ 100y0
+ 90y = 0. The discriminant of the characteristic
equation is greater than zero. So, it has two roots -9 and -1. The general solution is
y = c1e 9t
+ c2e t
:
By consider the initial conditions, the solution is
y =  0:02e 9t
+ 0:18e t
:yyyy

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