10.16 Solving Differential Equations Using Power Series

• A power series is an infinite series of the form \sum_{n = 0}^\infty a_{n} (x - c)^{n}
• What does $a_{n}$ represent in a power series?


  Coefficients
• The center of a power series is denoted by the variable c
• The coefficients in a power series determine its convergence and behavior.
• What formula is used to represent a function as a power series centered at c</latex>?

  Taylor series
• A Maclaurin series is a special case of a Taylor series centered at 0
• A Maclaurin series is best suited for approximating functions near $x =$$0$.
• Steps to find the Taylor series for f(x) = e^{x}</latex> centered at $c =$$1$


  1️⃣ Find the derivatives of $f(x)$.

  2️⃣ Evaluate the derivatives at $c$.

  3️⃣ Substitute into the Taylor series formula.
• What rule is applied to differentiate a power series term-by-term?

  Power rule
• The starting index of a differentiated power series changes to $n =$$1$ because the derivative of the constant term is zero.
• When differentiating $\sum_{n = 0}^\infty \frac{1}{n!} x^{n}$, the result is \sum_{n = 0}^\infty \frac{1}{n!} x^{n}
• Which type of function remains unchanged after differentiation?

  Exponential function
• The center of a power series shifts the series horizontally.
• The Maclaurin series for $e^{x}$ is \sum_{n = 0}^\infty \frac{x^{n}}{n!}
• Where is the Taylor series best approximated near?

  $x =$$c$
• The derivatives of $f(x) =$$e^{x}$ are all equal to $e^{x}$.
• What is the value of $f^{(n)}(1)$ for $f(x) =$$e^{x}$?


  $e$
• The Taylor series for $f(x) =$$e^{x}$ centered at $x =$$1$ is $\sum_{n = 0}^\infty \frac{e}{n!} (x - 1)^{n}$
• What rule is applied to differentiate a power series term-by-term?

  Power rule
• The starting index for the derivative of a power series changes to $n =$$1$ because the derivative of the constant term is zero.
• Steps to differentiate the power series $f(x) =$$\sum_{n = 0}^\infty \frac{1}{n!} x^{n}$


  1️⃣ Original series: $f(x) =$$\sum_{n = 0}^\infty \frac{1}{n!} x^{n}$

  2️⃣ Differentiate: $f'(x) =$$\sum_{n = 1}^\infty \frac{1}{(n - 1)!} x^{n - 1}$

  3️⃣ Simplify: $f'(x) =$$\sum_{n = 0}^\infty \frac{1}{n!} x^{n}$
• What is the simplified derivative of $f(x) =$$\sum_{n = 0}^\infty \frac{1}{n!} x^{n}$?


  $f'(x) =$$\sum_{n = 0}^\infty \frac{1}{n!} x^{n}$
• To solve a linear homogeneous differential equation using a power series, we assume a solution of the form y(x) = \sum_{n = 0}^\infty a_{n} x^{n}</latex> and substitute it into the equation.
• The first step to solving a differential equation using power series is to substitute the power series into the differential equation.
• Steps to solve a linear homogeneous differential equation using power series

  1️⃣ Substitute the power series into the differential equation

  2️⃣ Derive the recurrence relation for $a_{n}$

  3️⃣ Determine coefficients $a_{n}$ based on the recurrence relation
• What type of differential equations can be solved using power series?

  Linear homogeneous
• To solve a linear homogeneous differential equation using a power series, we assume a solution of the form y(x) = \sum_{n = 0}^\infty a_{n} x^{n}</latex>, where $a_{n}$ are the coefficients
• Steps to solve a linear homogeneous differential equation using a power series

  1️⃣ Substitute the power series into the differential equation

  2️⃣ Derive the recurrence relation for $a_{n}$

  3️⃣ Determine the coefficients $a_{n}$ based on the recurrence relation
• The example y'' + y = 0</latex> can be solved using a power series centered at $x =$$0$
• What is the recurrence relation for $a_{n + 2}$ in the example $y'' +$$y =$$0$?


  $a_{n + 2} =$$- \frac{a_{n}}{(n + 2)(n + 1)}$
• Match the index $n$ with its corresponding coefficient $a_{n}$ for the example $y'' +$$y =$$0$:


  0 ↔️ $a_{0}$

  1 ↔️ $a_{1}$

  2 ↔️ $- \frac{a_{0}}{2!}$

  3 ↔️ $- \frac{a_{1}}{3!}$
• What is the final solution to the example $y'' +$$y =$$0$?


  $y(x) =$$a_{0} \cos(x) +$$a_{1} \sin(x)$
• A linear non-homogeneous differential equation includes terms that are functions ofx</latex> but not $y$ or its derivatives
• To solve a non-homogeneous equation using a power series, we need to determine the coefficients $a_{n}$ based on a recurrence relation
• What are the initial conditions for the example $y'' +$$y =$$x$?


  $y(0) =$$0, y'(0) =$$1$
• The solution to $y'' +$$y =$$x$ with the given initial conditions is $y(x) =$$\sin(x) - \cos(x)$
• A power series is an infinite series of the form $\sum_{n = 0}^\infty a_{n} (x - c)^{n}$, where $c$ is the center of the series
• What does the center of a power series $c$ determine?


  Horizontal shift
• Match the component of a power series with its description:

  x ↔️ Variable

  aₙ ↔️ Coefficients

  c ↔️ Center of the series
• The coefficients of a power series determine its behavior and convergence