Final Review

Question 1

What is the main topic of Chapter 5?

Answer 1

Power Series

Question 2

What method is used to solve linear ODE in Chapter 5?

Answer 2

Power Series Method

Question 3

What is the assumed form of y in the power series method?

Answer 3

y = sum from n=0 to infinity a_n x^n

Question 4

What is the expression for y’ in the power series method?

Answer 4

y’ = sum from n=1 to infinity n a_n x^{n-1}

Question 5

What is the expression for y’’ in the power series method?

Answer 5

y’’ = sum from n=2 to infinity n(n-1) a_n x^{n-2}

Question 6

How do you solve for the recurrence relation in the power series method?

Answer 6

Solve for the highest term

Question 7

What is the general form of the recurrence relation?

Answer 7

a_{n+1} = f(n) a_n

Question 8

What types of singular points are mentioned?

Answer 8

Regular or irregular

Question 9

What is a Frobenius solution used for?

Answer 9

To find a solution at a regular singular point

Question 10

What is the assumed form of y in the Frobenius method?

Answer 10

y = sum from n=0 to infinity c_n x^{n+r}

Question 11

What is the expression for y’ in the Frobenius method?

Answer 11

y’ = sum from n=0 to infinity c_n (n+r) x^{n+r-1}

Question 12

What is the expression for y’’ in the Frobenius method?

Answer 12

y’’ = sum from n=0 to infinity c_n (n+r)(n+r-1) x^{n+r-2}

Question 13

What do you find after plugging into the ODE in the Frobenius method?

Answer 13

Indicial roots r

Question 14

What is the main topic of Chapter 12?

Answer 14

Fourier Series

Question 15

What is the interval for the Fourier series mentioned?

Answer 15

(-L, L)

Question 16

What is the general form of the Fourier series on (-L, L)?

Answer 16

f(x) = (1/2) a_0 + sum from n=1 to infinity [a_n cos(n pi x / L) + b_n sin(n pi x / L)]

Question 17

What is the formula for a_0 in the Fourier series?

Answer 17

a_0 = (1/L) integral from -L to L of f(x) dx

Question 18

What is the formula for a_n in the Fourier series for n >= 1?

Answer 18

a_n = (1/L) integral from -L to L of f(x) cos(n pi x / L) dx

Question 19

What is the formula for b_n in the Fourier series?

Answer 19

b_n = (1/L) integral from -L to L of f(x) sin(n pi x / L) dx

Question 20

What is the Odd-Even Theorem?

Answer 20

For odd f(x), a_n = 0; for even f(x), b_n = 0

Question 21

If f(x) is odd, what happens to the coefficients?

Answer 21

a_n = 0

Question 22

If f(x) is even, what happens to the coefficients?

Answer 22

b_n = 0

Question 23

What are half-range series?

Answer 23

Series on (0, L)

Question 24

What types of half-range series are there?

Answer 24

Fourier Sine Series and Fourier Cosine Series

Question 25

What is the main topic of Chapter 13?

Answer 25

Partial Differential Equations (PDEs)

Question 26

How do you form a product solution for PDEs?

Answer 26

By assuming the solution is separable

Question 27

What is the separable form for u(x,y)?

Answer 27

u(x,y) = v(x) w(y)

Question 28

In separation of variables, what equation is set?

Answer 28

f(x) = g(y) = - lambda, where lambda is the separation constant

Question 29

What cases are considered for the separation constant lambda?

Answer 29

lambda = 0, lambda < 0, lambda > 0

Question 30

What is the Heat Equation?

Answer 30

u_t = k u_xx

Question 31

What is the Wave Equation?

Answer 31

u_tt = a^2 u_xx

Question 32

What is Laplace's Equation?

Answer 32

u_xx + u_yy = 0