Variable Coefficient Equations

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

• Section 4.7 – Variable Coefficient Equations

Expected Educational Results

• Objective 14–1: I can identify variable-coefficient equations.
• Objective 14–2: I can find a solution to a homogeneous Cauchy-Euler equation.
• Objective 14–3: I can find a solution to a nonhomogeneous Cauchy-Euler equation.
• Objective 14–4: Given a solution, I can apply the Reduction of Order Theorem to find another solution to a Cauchy-Euler equation.
• Objective 14-5: I can find a solution to a Cauchy-Euler IVP.

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: $\displaystyle at^2y^{\,\prime\prime}(t)+bty^{\,\prime}(t)+cy(t)=0$, $t>0$.

The goal is to rewrite the ODE as $\displaystyle ay^{\,\prime\prime}(x)+by^{\,\prime}(x)+cy(x)=0$ by making a substitution. Explain why.

1. Let $t=e^x$.

2. Differentiate the let statement with respect to $x$.

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

4. Multiply the equation from Step 3 by $\displaystyle \frac{dy}{dt}$.

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 $t$.

7. Using the equation from Step 6:

   1. Multiply the side with the single term by $\displaystyle \frac{dt}{dx}$ and consider the derivatives as differentials and simplify.
   2. Multiply the other side by $\displaystyle t$.
   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 $\displaystyle at^2y^{\,\prime\prime}(t)+bty^{\,\prime}(t)+cy(t)=0$. Use Step 1 to show $y(t)=y(x)$. Now the DE is in terms of $y(x)$. Simplify.

12. Suppose $r_1$ and $r_2$ 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 $t$. Simplify. What do you notice?

14. Suppose $r_1$ 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 $r=\alpha\pm\beta i$ are complex roots to the characteristic equation for the simplified DE from Step 11. Repeat Step 12 and Step 13. What do you notice?

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