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Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

Geoffrey W. Cox, Ph.D., ?

Section 6.6 Summary & Exercises

The LHCC chapter (Linear Homogeneous Differential Equations with Constant Coefficients) in "Linear Constant Coefficient Methods" introduces a systematic method to solve higher-order linear differential equations. Here’s a summary based on the content:

Summary of the Key Ideas.

  • Linear Homogeneous Differential Equations with Constant Coefficients (LHCC)
    • These are differential equations where each term consists of a derivative of the unknown function multiplied by a constant.
    • The general form of an LHCC equation is:
      an y(n)+an−1 y(n−1)+⋯+a1 y′+a0 y=0.
  • The Characteristic Equation
    • By assuming a solution of the form y=erx, an LHCC can be reduced to a characteristic polynomial in r.
    • The solutions to the characteristic equation determine the form of the general solution.
  • Solution Types
    • Let r be a solution to the characteristic equation (CE).
    • If r is different from all other solutions of the CE, then
      cerx
      is a term of the general solution.
    • If r is equal to, say, three other solutions of the CE, then
      c1erx+c2xerx+c3x2erx
      are terms of the general solution.
    • If r=α+iβ or r=α−iβ, then the general solution contains
      eαx(c1sin⁡(βx)+c2cos⁡(βx)).

Method 4. LHCC Method.

The general solution to a linear homogeneous differential equation with constant coefficients (LHCC) of the form
(31)an y(n)+an−1 y(n−1)+⋯+a2 y″+a1 y′+a0 y=0,
can be found through the following steps...
Step 1: Solve the Characteristic Equation
Solve the characteristic equation (CE)
an rn+an−1 rn−1+⋯+a2 r2+a1 r+a0=0,
Step 2: Write Down the General Solution
  • Real & Different: r1,r2,…,rn
    y(x)=c1er1x+c2er2x+⋯+cnernx.
  • Real & Repeated: r1 (multiplicity m)
    y(x)=(c1+c2x+⋯+cmxm−1)er1x.
  • Complex: α±iβ
    y(x)=eαx(c1cos⁡(βx)+c2sin⁡(βx)).
  • For mixed root types, combine the corresponding terms to form the complete general solution.

Exercises Exercises

1.

Show why x is needed in the general solution for repeated roots of the CE
Answer. Answer

Classify the following differential equations as either homogeneous or nonhomogeneous.

2.

4y″−36y′=0
Answer. Answer

Find the general solution for each of the following.

3.

4y″−36y′=0
Answer. Answer

4.

4y″−36y=0.
Answer. Answer

5.

y″−y′−11y=0
Answer. Answer

6.

2d2θdt2−6dθdt−8θ=0
Answer. Answer

Solve the following initial value problems.

7.

4y″−36y=0c1 an c2 that satisfy thegiven initial conditions.
Answer. Answer
y=e3t+3e−3t o y=e3x+3e−3x

8.

2d2θdt2−6dθdt−8θ=0,θ(0)=12,θ′(0)=−2
Answer. Answer

Boundary Value Problems.

9.

Solve the following boundary value problem.y″−y=0,y(0)=1,y(1)=2e−1e
Answer. Answer
y=2et−e−t

Solve the following DEs.

10.

w″+6w′+9w=0
Answer. Answer

11.

m″=2m′−m
Answer. Answer

Initial Value Problems.

12.

Solve the following initial value problem.d2zdx2−4dzdx+4z=0,z(1)=1,z′(1)=1
Answer. Answer
z=(2e−2−e−2x)e2x or z=(2−x)e2x−2

Solve the following DEs.

13.

y″+4y′+53y=0
Answer. Answer

14.

z″=−36z
Answer. Answer

Solve the following DEs.

15.

y″=−24y′−144y
Answer. Answer

16.

d2wdx2−49w=0
Answer. Answer

17.

d2wdx2+49w=0
Answer. Answer

18.

z″−z′−42z=0
Answer. Answer

Find the general solution y(t) for a linear, homogeneous DEwith constant coefficients which has the given characteristic equation.

19.

(r−1)2(r+3)(r2+2r+5)2=0
Answer. Answer

20.

(r+1)2(r−6)3(r2+1)(r2+4)=0
Answer. Answer

Compute the derivative of each of the following functions.

21.

f(x)=ln⁡xcos⁡x
Answer. Answer
f′(x)=(1x)cos⁡x+ln⁡x(−sin⁡x)=cos⁡xx−ln⁡xsin⁡x

22. Classifying Practice.

    Select each classification label that applies to the equation
    y″=y′+6y
  • Linear
  • Correct, each of the terms are linear.
  • Homogeneous
  • Correct, the constant term is zero.
  • Constant Coefficients
  • Correct, each coefficient is constant.
  • LHCC
  • Correct!

23. Classifying Practice.

    Select each classification label that applies to the equation
    3y‴+y′−sin⁡(y)=0
  • Linear
  • Incorrect, sin⁡(y) is a nonlinear term.
  • Homogeneous
  • Technically, only linear equations can be labeled as homogeneous or not. Since the equation is nonlinear, we do not select it.
  • Constant Coefficients
  • Technically, only linear equations can be labeled as having constant coefficients or not. Since the equation is nonlinear, we do not select it.
  • LHCC
  • Incorrect.

24. Classifying Practice.

    Select each classification label that applies to the equation
    y″−6=0
  • Linear
  • Correct, both terms are linear.
  • Homogeneous
  • Incorrect, the constant term, 6, is non-zero.
  • Constant Coefficients
  • Correct, each coefficient is constant.
  • LHCC
  • Incorrect.

25. Classifying Practice.

    Select each classification label that applies to the equation
    d3ydt3+kdydt=ty,k is constant
  • Linear
  • Correct, all terms are linear.
  • Homogeneous
  • Correct, the constant term is zero.
  • Constant Coefficients
  • Incorrect, the y term coefficient, t, is not constant.
  • LHCC
  • Incorrect.
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