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Review

Euler: \[ y_{n+1} = y_n + f(t_n, y_n)h \]
Improved Euler: \[ y_{n+1} = y_n + \frac{1}{2} [f(t_n, y_n) + f(t_{n+1}, y_{n+1})]h \]

\( y_{n+1} \) on the right side is

Euler \( y_{n+1} = y_n + f(t_n, y_n)h \)

No RK4 on exam.

Sine / cosine series

\[ f(t) = t \]

\[ 0 < t < \pi \quad \text{period } 2\pi \]

cosine series \( \rightarrow \) add even extensions

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A coordinate graph showing the even extension of f(t)=t. The horizontal axis is t and the vertical axis is f(t). A solid V-shape is drawn from -pi to pi, passing through the origin. Dashed lines continue the periodic triangular wave pattern.

cosine series has no sine terms

\[ b_n = 0 \quad \text{for all } n \]

\[ a_n = \frac{2}{L} \int_{0}^{L} f(t) \cos\left(\frac{n\pi t}{L}\right) dt \]

here, \( L = \pi \)

\[ a_0 = \frac{2}{\pi} \underbrace{\int_{0}^{\pi} t \, dt}_{\text{area}} = \frac{2}{\pi} \left[ (\pi)(\pi)(\frac{1}{2}) \right] = \pi \]

\( \leftarrow \) avg. value of \( f(t) \) when in series \( (\frac{a_0}{2}) \)

\[ a_n = \frac{2}{\pi} \int_{0}^{\pi} t \cos(nt) \, dt \]

\[ u = t \quad dv = \cos(nt) \, dt \]

\[ du = dt \quad v = \frac{1}{n} \sin(nt) \]

\[ = \frac{2}{\pi} \left( \left. \frac{t}{n} \sin(nt) \right|_0^\pi - \frac{1}{n} \int_{0}^{\pi} \sin(nt) \, dt \right) \]

\[ = -\frac{2}{n\pi} \left( -\left. \frac{1}{n} \cos(nt) \right|_0^\pi \right) = \frac{2}{n^2\pi} \left[ (-1)^n - 1 \right] \]

\[ f(t) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nt) \]

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  • even · even is even
  • even · odd is odd
  • odd · odd is even

if \( f(t) \) even

\( f(t) \sin\left(\frac{n\pi t}{L}\right) \) is even · odd = odd

so \( b_n \neq 0 \) in general

\[ f(t) \sim \frac{1}{2} a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi t}{L}\right) \]
  • Converges to \( f(t) \) if \( f(t) \) is continuous at some \( t \)
  • Converges to \( \frac{f(t^-) + f(t^+)}{2} \) if \( f(t) \) is discontinuous at \( t \)
A graph showing a periodic function f(t) with linear segments. It has jump discontinuities at pi and 2 pi, with dots indicating the convergence point at the average of the jump.

converges to \( \frac{\pi}{2} \) at \( n\pi \)

converges to \( f(t) = t \) at other \( t \)

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Boundary-value problem

\[ X'' + kX = F(t) \]

\( X(0) = X(L) = 0 \)

\( X'(0) = X'(L) = 0 \)

with a given \( F(t) \) on \( 0 < t < L \)

  • we add even extensions if \( X'(0) = X'(L) = 0 \)
  • we add odd extensions if \( X(0) = X(L) = 0 \)
  • expand \( F(t) \)
  • assume \( X(t) \) is the same kind of series
  • plug into eq. to find coefficients

for example,

\[ X'' + 5X = F(t) \]

\( F(t) = 3 \quad 0 < t < \pi \)

\( X(0) = X(\pi) = 0 \)

\( X(0) = X(\pi) = 0 \) (positions fixed)

we expand \( F(t) = 3 \) as a sine series w/ period \( 2\pi \)

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Fourier Sine Series Expansion and Differential Equations

A coordinate graph showing a periodic square wave function F(t). The vertical axis is labeled F(t) with values 3 and -3. The horizontal axis is labeled t with values -pi, 0, and pi. The function is 3 between 0 and pi, and -3 between -pi and 0, forming a step-like pattern.

\( F(0) = F(\pi) = 0 \) (meets BC's)

Expansion of \( F(t) \)

Expand \( F(t) = 3 \) for \( 0 < t < \pi \) (where \( L = \pi \)) as a sine series.

\( a_n = 0 \) for all \( n \)

\[ b_n = \frac{2}{\pi} \int_{0}^{\pi} 3 \cdot \sin(nt) \, dt = \dots \]

\[ = \frac{6 [1 - (-1)^n]}{n\pi} \]

\[ F(t) = \sum_{n=1}^{\infty} \frac{6 [1 - (-1)^n]}{n\pi} \sin(nt) \]

Solving the Differential Equation

We assume a solution of the form \( x(t) = \sum_{n=1}^{\infty} B_n \sin(nt) \)

Sub into \( x'' + 5x = F(t) \):

\[ x'' + 5x = \sum_{n=1}^{\infty} \frac{6 [1 - (-1)^n]}{n\pi} \sin(nt) \]

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Derivatives and Substitution

\[ x' = \sum_{n=1}^{\infty} n B_n \cos(nt) \]

\[ x'' = \sum_{n=1}^{\infty} -n^2 B_n \sin(nt) \]

\[ \sum_{n=1}^{\infty} -n^2 B_n \sin(nt) + \sum_{n=1}^{\infty} 5 B_n \sin(nt) = \sum_{n=1}^{\infty} \frac{6 [1 - (-1)^n]}{n\pi} \sin(nt) \]

Solving for Coefficients

For each \( n \),

\[ -n^2 B_n + 5 B_n = \frac{6 [1 - (-1)^n]}{n\pi} \]

\[ B_n = \frac{6 [1 - (-1)^n]}{n\pi (5 - n^2)} \]

Final Solution

So, \( x(t) = \sum_{n=1}^{\infty} \frac{6 [1 - (-1)^n]}{n\pi (5 - n^2)} \sin(nt) \)

If same eq. but \( x'(0) = x'(\pi) = 0 \)

\( F(t) = 3 \)

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then even extension

A coordinate system with horizontal axis t and vertical axis F(t). A dashed horizontal line is drawn at height 3, extending from t = -pi to t = pi, illustrating an even extension of a function.

\( b_n = 0, \quad a_0 = 6, \quad a_n = 0 \quad n \ge 1 \)

still assume \( x(t) \) is a cosine series

\[ x(t) = \frac{1}{2} A_0 + \sum_{n=1}^{\infty} A_n \cos(nt) \]
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MA 30300-JC1/JC2
Exam 2
Spring 2026

Fourier Series and Differential Equations Reference

Term Formula Period
\( f(x) \) \( \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right) \) \( 2L \)
\( a_n \) \( \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx \) \( 2L \)
\( b_n \) \( \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \) \( 2L \)
Even Extension \( b_n = 0, \quad a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx \) \( 2L \)
Odd Extension \( a_n = 0, \quad b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \) \( 2L \)

Trigonometric Identities

  • \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \)
  • \( \cos(A)\cos(B) = \frac{1}{2}[\cos(A+B) + \cos(A-B)] \)
  • \( \cos(n\pi) = (-1)^n, \quad \sin(n\pi) = 0 \)

Integration

  • \( \int u \, dv = uv - \int v \, du \)

Common Trigonometric Integrals (for integer m, n)

  • \( \int_{0}^{\pi} \sin(nx) \cos(mx) \, dx = \begin{cases} \frac{2n}{n^2-m^2} & \text{if } n-m \text{ is odd} \\ 0 & \text{if } n-m \text{ is even} \end{cases} \)
  • \( \int_{-L}^{L} \cos\left(\frac{n\pi x}{L}\right) \cos\left(\frac{m\pi x}{L}\right) \, dx = \begin{cases} L & \text{if } n=m \neq 0 \\ 0 & \text{if } n \neq m \end{cases} \)
  • \( \int_{-L}^{L} \sin\left(\frac{n\pi x}{L}\right) \sin\left(\frac{m\pi x}{L}\right) \, dx = \begin{cases} L & \text{if } n=m \neq 0 \\ 0 & \text{if } n \neq m \end{cases} \)
  • \( \int_{-L}^{L} \sin\left(\frac{n\pi x}{L}\right) \cos\left(\frac{m\pi x}{L}\right) \, dx = 0 \text{ for all } n, m \)

Standard Form Integrals

  • \( \int x \cos(ax) \, dx = \frac{\cos(ax)}{a^2} + \frac{x \sin(ax)}{a} + C \)
  • \( \int x \sin(ax) \, dx = \frac{\sin(ax)}{a^2} - \frac{x \cos(ax)}{a} + C \)