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Section 2.8: The Existence and Uniqueness Theorem

Anintegral equation relates a function to its integral. To transform a first-order differential equation, we suppose temporarily that satisfies the initial value problem. Thus:

In the method of successive approximations,, we choose an initial function to approximate the solution, plug it into the integral equation, evaluate the integral to get a new function , and then use that new function to get a better approximate function, etc.:

…

Section 3.1: Homogeneous Equations with Constant Coefficients

Big idea: has solution or for some values r.

Here are four fundamental second-order differential equations and their solutions:

To solve a second-order differential equation with constant coefficients:

Assume a solution of the form , substitute it and its derivatives and into the equation.

Then find the values r1 and r2 that satisfy the resulting characteristic equation:

The general solution is a linear combination of the two solutions:

Section 3.2: Solutions of Linear Homogeneous Equations; the Wronskian

Big Idea:Any two solutions we find of a homogeneous linear second-order differential equation are all that is needed if the Wronskian is non-zero on the interval of the existence of the solution.

Theorem 3.2.1: Existence and Uniqueness Theorem

(for Linear Second-Order Differential Equations)

For the initial value problem

where p, q, and g are continuous on some open interval I containing the point t0, then there is exactly one solution of this problem, and the solution exists throughout the interval I.

Theorem 3.2.2: Principle of Superposition

If y1 and y2 are solutions of the differential equation

,

then the linear combination c1y1 + c2y2 is a solution for any values of the constants c1 and c2.

The Wronskian:

Theorem 3.2.3:

If y1 and y2 are two solutions of the differential equation

and the initial conditions must be satisfied by y, then it is always possible to choose the constants c1 and c2 so that

satisfies this initial value problem if and only if the Wronskianis not zero at t0.

Theorem 3.2.4:

If y1 and y2 are two solutions of the differential equation

then the family of solutions

with arbitrary constants c1 and c2 includes every solution of the differential equation if and only if there is a point t0 where the Wronskianis not zero.

Notes:

  • is called the general solution of .
  • y1 and y2 are said to form a fundamental set of solutions.
  • To find the general solution, and thus all solutions of , we need only find two solutions of the given equation whose Wronskian is non-zero.

Theorem 3.2.5:

If the differential equation

has coefficients p and q that are continuous on some open interval I, and if t0I, and if y1 is a solution that satisfies the initial conditions

and if y2 is a solution that satisfies the initial conditions

then y1 and y2 form a fundamental set of solutions of the equation.

Theorem 3.2.6: (Abel’s Theorem)

If y1 and y2 are two solutions of the differential equation

With the coefficients p and q continuous on an open interval I, then the Wronskian is given by:

where c is a certain constant that depends on y1andy2, but not on t. Further, either is zero for all tI (if c = 0), or else is never zero in I (if c 0).

Section 3.3: Complex Roots of the Characteristic Equation

Big Idea: When there are complex roots to the characteristic equation, each of the fundamental solutions will have a sinusoidal factor.

Euler’s Formula:

Notes:

  • The complex solutions from the quadratic formula always come in conjugate pairs.
  • If those complex roots are , then the general solution can always be written as .

Additional topics from homework:

  • A Euler equation is of the form . It can be solved by making the substitution , which results in the constant coefficient equation
  • A general linear homogeneous equation can be solved using the substitution if and only if .

Section 3.4: Repeated Roots; Reduction of Order

Big Idea: When there are repeated roots to the characteristic equation, the second of the fundamental solutions is formed by multiplying the first solution by the independent variable.

Summary of Sections 3.1 – 3.4:

  • To solve a second-order linear homogeneous differential equation with constant coefficients:
  • Assume a solution of the form , which leads to a corresponding characteristic equation:
  • Solve the characteristic equation to find its two roots r1 and r2:
  • If the roots are real and unequal, then the general solution is
  • If the roots are complex conjugates, then the general solution is
  • If the roots are the same, then the general solution is
  • If we have a more complicated second-order initial value problem like: , and we know that y1 and y2 are solutions of the differential equation, then we can be sure that they form a fundamental set of solutions if the Wronskian is nonzero:
  • Reduction of Order (end of 3.4):If we know a solution y1 of the more general second-order differential equation: , then we can derive the second solution y2 by assuming and substitutey2 into the original equation, resulting in , which is a first order equation for v.

Section 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients

Big idea: One way to solve is to “guess” a particular solution that has the same form as g(t), then work out the value(s) of the coefficient(s) that make that guess work.

Theorem 3.5.1: The Difference of Nonhomogeneous Solutions Is the Homogeneous Solution

If Y1 and Y2 are two solutions of the nonhomogeneous equation , then their difference is a solution of the corresponding homogeneous solution . If, in addition, y1 and y2 are a fundamental set of solutions of the homogeneous equation, then

,

for certain constants c1 and c2.

Theorem 3.5.2: General Solution of a Nonhomogeneous Equation

The general solution of the nonhomogeneous equation , can be written in the form m where , y1 and y2 are a fundamental set of solutions of the corresponding homogeneous equation, c1 and c2, are arbitrary constants, and Y is some specific solution of the nonhomogeneous equation.

Method of Undetermined Coefficients:

When g(t) is an exponential, sinusoid, or polynomial, make a guess for the particular solution that is the same exponential, sinusoid, or polynomial, except with arbitrary coefficients that you determine by substituting your guess into the nonhomogeneous equation.

The Particular Solution of
/ ; s is the smallest nonnegative integer so that no term replicates any term in the homogeneous equation.
/ ; s is the smallest nonnegative integer so that no term replicates any term in the homogeneous equation.

Section 3.6: Variation of Parameters

Big Idea:W you know the two homogeneous solutions, use . It assumes a solution that is of the form of a sum of the products of arbitrary functions and the fundamental homogeneous solutions.

Big Skill: You should be able to apply the technique of variation of parameters to solve nonhomogeneous equations.

The basic idea behind the method of variation of parameters, due to Lagrange, is to find two fundamental solutions y1 and y2 of a corresponding homogeneous equation, then assume that the solution of the nonhomogeneous equation is:

with the additional criterion that .

Practice:

  1. Find the general solution of the equation using the method of variation of parameters. Verify that the formula in theorem 3.6.1 produces the same solution

  1. Find the general solution of the equation using the method of variation of parameters, using .

Theorem 3.6.1: Solution of a Nonhomogeneous Second-Order Equation

If the functions p, q, and g are continuous on an open interval I, and if the functions y1 and y2 are a fundamental set of solutions of the homogeneous equation corresponding to the nonhomogeneous equation , then a particular solution is:

,

where t0 is any conveniently chosen point in I. The general solution is

.

Practice:

  1. Prove theorem 3.6.1, if there is time. If not, read pages 187 and 188…

Section 3.7: Mechanical and Electrical Vibrations

Big Idea: Second order differential equations with constant coefficients provide very good models for oscillating systems, like a mass hanging from a spring or a series RLC circuit.

Big Skill: You should be able to solve oscillating system IVPs.

Practice:

  1. Draw a free body diagram, and then write out a differential equation to describe the motion of the mass.

  1. State the units on all constants in MKS, cgs, and English units of measurement.

  1. Solve the equation when the damping is zero.

  1. Solve an undamped IVP.

  1. Solve the equation when the damping is not zero.Since damping slows things down, the period of oscillation is increased, which means the frequency is decreased. Find the “quasiperiod” and “quasifrequency” of the system.

  1. Solve an undamped IVP…

  1. Look at critically and overdamped systems.

  1. Extend the analog to a series RLC Circuit…

Section 3.8: Forced Vibrations

Big Idea: A damped spring-mass (or circuit) system described in section 3.7 that is driven by a sinusoidal force will eventually settle into a steady oscillation with the same frequency as the driving force. The amplitude of this steady state solution increases as the drive frequency gets closer to the natural frequency of the system, and as the damping decreases. An undamped spring-mass system that is driven by a sinusoidal force results in an oscillation that is the average of the two frequencies and modulated by a sinusoidal envelope that is the difference of the frequencies.

Big Skill: You should be able to solve sinusoidally-forced vibration problems.

Problems:

  1. Find the solution of a spring-mass system IVP

Forced Vibrations with Damping

In this section, we will restrict our discussion to the case where the forcing function is a sinusoid. Thus, we can make some general statements about the solution:

The equation of motion with damping will be given by:

Its solution will be of the form:

Notes:

  • The homogeneous solution , which is why it is called the “transient solution.”
  • The constants c1 and c2 of the transient solution are used to satisfy given initial conditions.
  • The particular solution is all that remains after the transient solution dies away, and is a steady oscillation at the same frequency of the driving function. That is why it is called the “steady state solution,” or the “forced response.”
  • The coefficients A and B must be determined by substitution into the differential equation.
  • If we replace with , then , , , , and . (See scanned notes at end for derivation)
  • Note that as , .
  • Note that when ,
  • Note that as , (mass is out of phase with drive).
  • The amplitude of the steady state solution can be written as a function of all the parameters of the system:
  • Notice that is dimensionless (but proportional to the amplitude of the motion), since is the distance a force of F0 would stretch a spring with spring constant k.
  • Notice that is dimensionless…
  • Note that as , .
  • Note that as , (i.e., the drive is so fast that the system cannot respond to it and so it remains stationary).
  • The frequency that generates the largest amplitude response is:

  • Plugging this value of the frequency into the amplitude formula gives us:
  • If , then the maximum value of R occurs for .
  • Resonance is the name for the phenomenon when the amplitude grows very large because the damping is relatively small and the drive frequency is close to the undriven frequency of oscillation of the system.

Practice:

  1. Find the solution of . Make graphs of the solution for various values of .


Demonstration:

  1. Compute the spring constant and resonant frequency of a given spring-mass system, and then verify the resonant frequency calculation experimentally. Observe the following:
  2. pulling up and down on the spring very rapidly results in virtually no motion of the mass.
  3. pulling up and down slowly results in the entire system simply moving up and down with the pull.
  4. pulling up and down near the resonant frequency results in a large oscillation for a pulling motion that is almost imperceptible to the human eye.

Forced Vibrations Without Damping

The equation of motion of an undamped forced oscillator is:

When (non-resonant case), the solution is of the form:

When (resonant case), the solution is of the form:

Practice:

  1. Derive both particular solutions above.

  1. Show that for the non-resonant case with initial condition (i.e., starting from rest at the equilibrium position), , and that the solution becomes , which can be written as using the sum-to-product trig identity .

Notice in the last problem that the solution can be considered to be a rapid-frequency oscillation that is modulated by a low-frequency oscillation. In electronics, this is called amplitude modulation. In acoustics, the low-frequency oscillation is called a beat because the sound wave is perceived as a constant pitch whose loudness varies with a beat.

  1. Solve the IVP

  1. Solve the IVP

Section 4.1: General Theory of nth Order Linear Equations

Big Idea: The techniques and theorems regarding second-order linear differential equations can be extended to higher-order differential equations.

Big Skill: You should be able to verify solutions, compute Wronskians, and determine linear independence of solutions.

Usual form of an nth order linear differential equation:

Assumptions:

  • The functionsare continuous and real-valued on some interval , and for any .

Linear differential operator form:

Notes:

  • Solving an nth order equation ostensibly requires n integrations
  • This implies n constants of integration
  • Also implies n initial conditions to completely specify an IVP:

Theorem 4.1.1: Existence and Uniqueness Theorem

(for nth-Order Linear Differential Equations)

If the functions where are continuous on the open interval I, then exists exactly one solution of the differential equation that also satisfied the initial conditions . This solution exists throughout the interval I.

Practice:

  1. Determine an interval in which the solution of is sure to exist.

The Homogeneous Equation:

Notes:

  • If the functions are solutions, then so is a linear combination of them,
  • To satisfy the initial conditions, we get n equations in n unknowns:
  • This system will have a solution for provided the determinant of the matrix of coefficients is not zero (i.e., Cramer’s Rule again). In other words, the Wronskian is nonzero, just like for second-order equations.
  • Note that a slightly modified form of Abel’s Theorem still applies:

Theorem 4.1.2:

If the functions where are continuous on the open interval I, if the functions are solutions of , and if for at least one point in I, then every solution of the differential equation can be written as a linear combination of .

Notes:

  • is called the general solution of .
  • are said to form a fundamental set of solutions.

Practice:

  1. Verify that are solutions of , and compute their Wronskian.

Linear Dependence and Independence:

  • The functions are said to be linearly dependent on an interval I if there exist constants , NOT ALL ZERO, such that FOR ALL.
  • The functions are said to be linearly independent on I if they are not linearly dependent there.

Practice:

  1. Determine if are linearly dependent or independent on . If they are dependent, write a linear relationship between them.

  1. Determine if are linearly dependent or independent on . If they are dependent, write a linear relationship between them.

Theorem 4.1.3:

If is a fundamental set of solutions to on an interval I, then are linearly independent on I. Conversely, if are linearly independent solutions of the equation, then they form a fundamental set of solutions on I.

The Nonhomogeneous Equation:

Notes:

  • If Y1(t) and Y2(t) are solutions of the nonhomogeneous equation, then
  • I.e., the difference of any two solutions of the nonhomogeneous equation is a solution of the homogeneous equation.
  • So, the general solution of the nonhomogeneous equation is: , where Y(t) is a particular solution of the nonhomogeneous equation.
  • We will see that the methods of undetermined coefficients and reduction of order can be extended from second-order equations to nth-order equations.

Section 4.2: Homogeneous Equations with Constant Coefficients

Big Idea: Homogeneous equations with constant coefficients can be solved by assuming a solution of the for ert, and then solving the consequent characteristic equation for r.

Big Skill: You should be able to solve these kinds of equations by finding roots of the characteristic equation.

For equations of the form

, “it is natural to anticipate that”
y = ert is a solution for correct values of r. Under this anticipation,

, where the polynomial

is called the characteristic polynomial, and is called the characteristic equation.

Recall that a polynomial of degree n has n zeros, and thus the characteristic polynomial can be written as:

Practice: All roots are real and unequal…

  1. Find the solution of the IVP .

Practice: Some roots are complex…

  • Recall that if a polynomial has real coefficients, then it can be factored into linear and irreducible quadratic factors.
  • The linear irreducible quadratic factors will factor into complex conjugate roots , which will correspond to solutions of .
  1. Find the solution of the IVP .

Practice: Some roots are repeated…

  • If a root r1is repeated s times, then that repeated root will generate s solutions:
  • The same applies if the repeated roots are complex.
  1. Find the general solution of .

  1. Find the general solution of .

Section 4.3: The Method of Undetermined Coefficients

Big Idea: The method of undetermined coefficients works for nth order equations just as it does for second-order equations.

Big Skill: You should be able to use the method of undetermined coefficients to find the particular solution of certain nonhomogeneous equations.

If the nonhomogeneous term g(t) of the linear nth order differential equation with constant coefficients

is of an “appropriate form” (i.e., a sinusoidal, polynomial, or exponential function), then the method of undetermined coefficients can be used to find the particular solution.

Recall that if any term of the proposed particular solution replicates a term of the homogeneous solution, then the entire particular solution must be multiplied by a sufficient number of factors of t to eliminate the replication.