Detailed Examples of Power Series Applied to Differential Equations

📚 Quick Study Guide

• 🔢 Power series are infinite series of the form $\sum_{n=0}^{\infty} a_n (x-c)^n$, where $a_n$ are coefficients and $c$ is the center.
• 🔍 When applying power series to differential equations, assume a solution of the form $y(x) = \sum_{n=0}^{\infty} a_n x^n$.
• 📝 Differentiate the assumed solution term by term to find $y'(x)$ and $y''(x)$.
• ➕ Substitute the power series representations of $y(x)$, $y'(x)$, and $y''(x)$ into the differential equation.
• ⚖️ Equate coefficients of like powers of $x$ to find recurrence relations for the coefficients $a_n$.
• 💡 Use the recurrence relation to find a pattern for the coefficients and express the solution in terms of known functions if possible.
• 🧪 Check the interval of convergence for the power series solution.

🧪 Practice Quiz

1. What is the general form of a power series centered at $x=0$?

   1. $ \sum_{n=1}^{\infty} a_n x^n $
   2. $ \sum_{n=0}^{\infty} a_n $
   3. $ \sum_{n=0}^{\infty} a_n x^n $
   4. $ \sum_{n=-\infty}^{\infty} a_n x^n $

2. Given the differential equation $y' + y = 0$, and assuming a solution $y(x) = \sum_{n=0}^{\infty} a_n x^n$, what is the recurrence relation?

   1. $a_{n+1} = a_n$
   2. $a_{n+1} = -a_n$
   3. $a_{n+1} = n a_n$
   4. $a_{n+1} = -n a_n$

3. What is the first step in applying power series to solve a differential equation?

   1. Find the roots of the characteristic equation.
   2. Assume a power series solution.
   3. Calculate the Laplace transform.
   4. Solve for the eigenvalues.

4. For the differential equation $y'' - x y = 0$, what substitution is used to find the power series solution?

   1. $y = e^{rx}$
   2. $y = \sum_{n=0}^{\infty} a_n x^n$
   3. $y = A \cos(x) + B \sin(x)$
   4. $y = x^r$

5. If $y(x) = \sum_{n=0}^{\infty} a_n x^n$, what is $y'(x)$?

   1. $ \sum_{n=1}^{\infty} n a_n x^{n-1} $
   2. $ \sum_{n=0}^{\infty} n a_n x^{n-1} $
   3. $ \sum_{n=0}^{\infty} a_n x^{n+1} $
   4. $ \sum_{n=1}^{\infty} a_n x^{n+1} $

6. In the power series method, what do you do after substituting the series into the differential equation?

   1. Solve for the roots.
   2. Equate coefficients of like powers of x.
   3. Integrate term by term.
   4. Apply initial conditions.

7. What is the interval of convergence for a power series solution extremely important?

   1. It determines the color of the graph.
   2. It defines where the solution is valid.
   3. It simplifies the solution process.
   4. It is not important.