Mock Exam
1. Answer the following questions.
1.1. Γ 3
2
=. . . .Γ(n) = . . . .(n is a positive integer).
1.2. The general solution of x2y00+xy0+ (x2−5)y= 0 is y=. . . . 1.3. The general solution of (1−x2)y00−2xy0+ 6y= 0 is y=. . . . 2. Use thepower series method to find the series solution y=P∞
n=0bnxn of the equation y00+xy0+y= 0.
Calculate the first 6 non-zero coefficients bn’s.
3. Use the Frobenius method to find a nonzero series solution y = xrP∞
n=0bnxn of the equation
4x2y00+ 4xy0−y = 0 (x >0).
4. Evaluate the following Laplace transforms 4.1. L[2 sin 3t]. . . ,L[√
t] =. . . . 4.2. L
e−tt
4.3. Letf(t) =
0 if 0< t <1 6 if 1< t <4 1 t >4
.
(a) Express f(t) in terms of the Heaviside function.
(b) Evaluate L[f(t)].
5. Evaluate the following inverse Laplace transforms 5.1. L−1
3 s+ 2
=. . . ,L−1 5s
s2+ 1
=. . . .
5.2. L−1
e−5s s2+ 9
5.3. L−1
s−3 s2−6s+ 10
6. Use the Laplace transformation method to solve the initial value problem:
(y00−y=δ(t−3), y(0) = 0, y0(0) = 0.
7. Let
f(x) = sin2(x).
Methods App Math 2301317, ISE Program 2 7.1. Show thatf(x) is 2π-periodic.
7.2. Find the Fourier series expansion of f(x).
8. Let h: [0,4]→Rbe a function defined by h(x) =
(0 if 0≤x <2,
−3 if 2< x≤4.
8.1. What is the odd extension ofh(x)?
8.2. Expand theFourier sine series of h(x).
9. Find the Fourier integral expansion of the function u(x) =
(2 |x|<3 0 |x|>3.
10. Find the Fourier transform of the function v(x) =
(e−4x if x >0 0 if x <0.
11. Solve the IBVP
utt = 4uxx 0< x < π, t >0,
u(0, t) = u(π, t) = 0 t >0,
u(x,0) =f(x), ∂u
∂t(x,0) = 0 0< x < π.
12. Solve the BVP
uxx+uyy = 0 0< x <1,0< y < π, ux(x,0) = ux(x, π) = 0 0< x <1,
u(0, y) = 0, u(1, y) = y 0< y < 1.
