Variable Coefficient Equations

Variable Coefficient EquationsExpected Educational ResultsVariable Coefficient EquationsReduction of OrderTheorem: Reduction of OrderExample 03:Method of Solving Nonhomogeneous Cauchy-Euler EquationsInvestigation 08Investigation 09CC BY-NC-SA 4.0

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

Expected Educational Results

Variable Coefficient Equations

Reduction of Order

Theorem: Reduction of Order

Let be a solution to the homogeneous DE on some interval. Then, , where is the coefficient of the first-order derivative term, is a second, linearly independent solution.

Example 03:

Consider . Since the second-order derivative does not have a coefficient with , this is not a Cauchy-Euler equation.

Verify is a solution. Find .

Solution:

So, . Thus, is a solution to the ode.

Since .

Let's find . By the Reduction of Order definition:

Thus, .

Method of Solving Nonhomogeneous Cauchy-Euler Equations

There are two options:

Investigation 08

Solve the following nonhomogeneous DEs:

  1. , [Hint: Use a substitution .]
  2. , .

Investigation 09

Solve the following homogeneous DEs:

  1. Suppose is a solution to , . Find the general solution.
  2. Suppose is a solution to , . Find the general solution.

CC BY-NC-SA 4.0

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Last Modified: Wednesday, 14 October 2020 2:48 EDT