Variable Coefficient Equations

Variable Coefficient EquationsExpected Educational ResultsVariable Coefficient EquationsMethod of Solving Homogeneous Cauchy-Euler EquationsInvestigation 03Investigation 04Investigation 05CC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Variable Coefficient Equations

Method of Solving Homogeneous Cauchy-Euler Equations

Let .

  1. From Activity 02, the characteristic equation is .

  2. Find the solutions to the characteristic equation, and .

    • If , then the linearly independent solutions are and and the homogeneous solution is
    • If , then the linearly independent solutions are and and the homogeneous solution is
    • If and are complex numbers in the form , then the linearly independent solutions are and and the homogeneous solution is

Investigation 03

  1. For what values of is not a solution to ? Explain.
  2. For what values of is a solution to ? Explain.
  3. Suppose is a solution to . Find the associated characteristic equation.

Example 01: Solve ,

Solution:

Note that the coefficients of each -term is a polynomial function of the independent variable in which the degree of the polynomial term is equivalent to the degree of the derivative of . Thus, this ode is a Cauchy-Euler equation.

Identify the coefficients from the ode: , , . and .

The, characteristic equation for the above Cauchy-Euler equation is: . Using Mathematica, the roots of the characteristic equation are: , where and .

Thus, the homogeneous solution is .

Investigation 04

  1. Use the Wronskian to verify and , where , are linearly independent. Explain.
  2. Why is the condition, , above necessary? Explain.
  3. Use the Wronskian to verify and are linearly independent. Explain.

Investigation 05

Solve the following homogeneous DEs:

  1. , .
  2. , .
  3. , .
  4. , .
  5. , . Hint: Let .

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Last Modified: Wednesday, 14 October 2020 11:42 EDT