Variable Coefficient Equations

Variable Coefficient EquationsExpected Educational ResultsVariable Coefficient EquationsDerivation of the Solutions to Cauchy-Euler EquationsActivity 02CC BY-NC-SA 4.0

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

Expected Educational Results

Variable Coefficient Equations

Derivation of the Solutions to Cauchy-Euler Equations

NOTE: You will not be assessed on the derivation of the solutions to the Cauchy-Euler Equations.

Activity 02

Given the homogeneous Cauchy-Euler equation: , .

The goal is to rewrite the ODE as by making a substitution. Explain why.

  1. Let .

  2. Differentiate the let statement with respect to .

  3. Rewrite equation from Step 2 using the let statement.

  4. Multiply the equation from Step 3 by .

  5. Consider the derivatives as differentials and simplify the side that contains product of the two derivatives.

  6. Differentiate the equation from Step 5 with respect to .

  7. Using the equation from Step 6:

    1. Multiply the side with the single term by and consider the derivatives as differentials and simplify.
    2. Multiply the other side by .
    3. Explain why Step 7-1 and Step 7-2 are valid.
  8. Rewrite the side of the equation from Step 7 that has two terms using the result from Step 5.

  9. Rewrite as a second derivative the side of the equation from Step 8 that has a single term.

  10. Solve the equation from Step 9 for the second derivative that has a coefficient.

  11. Substitute the equations from Step 5 and Step 10 into . Use Step 1 to show . Now the DE is in terms of . Simplify.

  12. Suppose and are the roots to the characteristic equation for the simplified DE from Step 11. Use the Method of Undetermined Coefficients to write the solutions to the DE.

  13. Use the let statement to rewrite the solutions Step 12 in terms of . Simplify. What do you notice?

  14. Suppose is a repeated root to the characteristic equation for the simplified DE from Step 11. Repeat Step 12 and Step 13. What do you notice?

  15. Suppose are complex roots to the characteristic equation for the simplified DE from Step 11. Repeat Step 12 and Step 13. What do you notice?

CC BY-NC-SA 4.0

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Last Modified: Wednesday, 14 October 2020 11:40 EDT