10.16 Solving Differential Equations Using Power Series

• What is a power series in the context of calculus?

  Infinite series with powers
• The general form of a power series is \sum_{n=0}^{\infty} c_n (x-a)^n
• The Maclaurin series for $e^{x}$ has a convergence interval of $( - \infty, \infty)$.


  True
• Steps to represent a function as a power series:

  1️⃣ Write the general form of a power series

  2️⃣ Determine the coefficients and center

  3️⃣ Check the convergence interval
• What is the coefficient for the term $x^{2n}$ in the Taylor series for $\cos x$ around a = 0</latex>?


  $\frac{( - 1)^{n}}{(2n)!}$
• Different power series examples show that coefficients and centers can create various series, each with its own convergence interval
• Match the power series with its coefficients, center, and convergence interval:

  Maclaurin Series for $e^{x}$ ↔️ Coefficients: $\frac{1}{n!}$, Center: 0, Convergence Interval: $( - \infty, \infty)$

  Taylor Series for $\sin x$ around $a =$$0$ ↔️ Coefficients: $\frac{( - 1)^{n}}{(2n + 1)!}$, Center: 0, Convergence Interval: $( - \infty, \infty)$

  Taylor Series for $\cos x$ around $a =$$0$ ↔️ Coefficients: $\frac{( - 1)^{n}}{(2n)!}$, Center: 0, Convergence Interval: $( - \infty, \infty)$

  Geometric Series with $r =$$x$ around $a =$$0$ ↔️ Coefficients: 1, Center: 0, Convergence Interval: $( - 1, 1)$
• Different coefficients and centers can create various power series, each with its own convergence interval
• The Taylor series for $\sin x$ around $a =$$0$ has a convergence interval of $( - 1, 1)$.


  False
• The geometric series with $r =$$x$ around $a =$$0$ converges for $|x| < 1$.


  True
• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$
• What is the Maclaurin series for $e^{x}$?


  \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}</latex>
• The geometric series with $r =$$x$ has a convergence interval of ( - 1, 1)
• What is the convergence interval for the Maclaurin series of $e^{x}$?


  $( - \infty, \infty)$
• A power series can be differentiated term by term within its convergence interval.
• What is the derivative of the power series for $e^{x}$?


  $\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
• What is the integral of the Maclaurin series for $e^{x}$?


  $\sum_{n = 0}^{\infty} \frac{x^{n + 1}}{(n + 1)!} +$$C$
• Match the function with its Maclaurin series:

  $e^{x}$ ↔️ $\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$

  $\sin x$ ↔️ $\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n + 1}}{(2n + 1)!}$
• A special case of a power series, the Maclaurin series, is centered at 0
• What is the center of a power series?

  $a$
• The convergence interval for $e^{x}$ is $( - \infty, \infty)$.


  True
• The convergence interval for \frac{1}{1 - x}</latex> is ( - 1, 1)
• Term-by-term differentiation of a power series is valid within its convergence interval.


  True
• The constant of integration in term-by-term integration of a power series is denoted by C
• The integral of $e^{x}$ represented as a power series is $e^{x} +$$C$.


  True
• When solving a differential equation using power series, the differential equation is transformed into an algebraic equation involving the coefficients
• Equating coefficients in a differential equation results in recurrence relations.


  True
• In recurrence relations, coefficients are expressed in terms of previous coefficients.
• The simplification step in equating coefficients involves combining terms on both sides to have equal powers of $x$.


  True
• Steps in equating coefficients to find recurrence relations

  1️⃣ Substitute power series for $y$ and its derivatives

  2️⃣ Simplify the equation

  3️⃣ Equate coefficients of like powers of $x$

  4️⃣ Define the recurrence relation
• Re-indexing the first sum in the example involves shifting the starting value of n.
• What are the initial conditions given in the example?

  $y(0) =$$1$ and $y'(0) =$$0$
• What is the power series solution for the differential equation in the example?

  $\cos x$
• Equating coefficients involves comparing coefficients of like powers of x
• What does a recurrence relation define in terms of coefficients?

  Previous coefficients
• Re-indexing the first sum in the differential equation requires starting at n = 0
• In the differential equation $y'' +$$y =$$0$ with initial conditions $y(0) =$$1$ and $y'(0) =$$0$, all odd terms in the power series are zero.


  True
• What is the general formula for the recurrence relation in the differential equation y'' + y = 0</latex>?

  $c_{n + 2} =$$- \frac{c_{n}}{(n + 2)(n + 1)}$
• Steps to solve a differential equation using a power series

  1️⃣ Set up the recurrence relation

  2️⃣ Solve the recurrence relation

  3️⃣ Substitute the coefficients
• What is the recurrence relation for the differential equation $y'' +$$y =$$0$?


  c_{n + 2} = - \frac{c_{n}}{(n + 2)(n + 1)}</latex>