1. HW 6, Part I
(1) Find the characteristic polynomials of the following second order equations and solve these equations.
(a) y00− 6y0+ 8y = 0.
(b) y00− 16y0+ 64y = 0.
(c) 4y00+ y = 0.
(2) Let us consider the following second order equation ψ00+ P (x)ψ0+ Q(x)ψ = 0.
Assume that y = y(x) and z = z(x) are solutions to the equation satisfying the initial conditions
y(0) z(0) y0(0) z0(0)
=1 0 0 1
Let us denote W (x) to be the determinant:
W (x) =
y(x) z(x) y0(x) z0(x) .
Show that W (x) satisfies the initial value problem (the following first order equation with initial condition is called an initial value problem)
W0+ P (x)W = 0, W (0) = 1.
(3) Let (sn) be the sequence of numbers defined by sn=
n
X
k=1
n n2+ k2. Compute lim
n→∞sn if it exists.
(4) Suppose that a1, · · · , an are real numbers. Denote Qn
k=1ak = a1 · a2· · · an. Let us define (xn) by
xn=
n
Y
k=1
1 +k
n
n1 . Compute lim
n→∞xn. (Hint: consider ln xn.) (5) Compute lim
n→∞
Z 1 0
xn 1 +√
xdx. (Hint: 0 ≤ xn 1 +√
x ≤ xn for 0 ≤ x ≤ 1.) (6) Use the integral
Z 1 0
(1 − x)ndx to show that
n 0
− 1 2
n 1
+ · · · +(−1)n n + 1
n n
= 1
n + 1.
(7) Let −∞ < a < b < ∞. Assume that y = f (x) and y = g(x) are continuous functions on [a, b].
(a) Assume that y = f (x) is not a zero function. Define F (t) =
Z b a
(tf (x) − g(x))2dx, t ∈ R.
Find A, B, C such that F (t) = At2− 2Bt + C.
(b) Explain that F (t) ≥ 0.
(c) Using (b), show that B2≤ AC.
1
2
(8) Let y = f (x) be a continuous function on [0, 2π]. Denote an= 1
π Z 2π
0
f (x) cos nxdx, n ≥ 0,
bn= 1 π
Z 2π 0
f (x) sin nxdx, n ≥ 1.
The infinite series a0
2 +
∞
X
n=1
{ancos nx + bnsin nx}
is called the Fourier expansion for y = f (x).
(a) f (x) = x, 0 ≤ x ≤ 2π. Compute the Fourier expansion for f.
(b) f (x) = x2, 0 ≤ x ≤ 2π. Compute the Fourier expansion for f.
In this exercise, you need to use the integration by parts formula:
Z b a
u(x)v0(x)dx = u(x)v(x)|bx=a− Z b
a
v(x)u0(x)dx.
(c) The following equality is called the Parseval’s identity:
1 π
Z 2π 0
|f (x)|2dx = a20 2 +
∞
X
n=1
{a2n+ b2n}.
Use (a) and the Parseval identity to compute ζ(2) =
∞
X
n=1
1 n2.
Similarly, using (b) and the Parseval identity to compute ζ(4) =
∞
X
n=1
1 n4.
(d) Suppose we know cos 3x = 4 cos3x − 3 cos x and sin 3x = 3 sin x − 4 sin3x. Using Parseval identity to compute
Z 2π 0
cos6xdx and Z 2π
0
sin6xdx.
(9) A function y = f (x) defined on R is a periodic function of period T if f (x+T ) = f (x) for all x ∈ R. Assume that y = f (x) is a continuous function on R of period T. For example, sin x and cos x are periodic functions of period 2π on R. Show that
Z a+T a
f (x)dx = Z T
0
f (x)dx, ∀a ∈ R.
(10) A function y = f (x) on R is even if f (x) = f (−x) for all x and is odd if f (x) = −f (x) for all x. For example, sin x is an odd function on R while cos x is even on R. Show that if y = f (x) is an even function, then
Z a
−a
f (x)dx = 2 Z a
0
f (x)dx and if y = g(x) is odd, then
Z a
−a
g(x)dx = 0.
3
Compute Z π
−π
sin2013x 4 + cos27xdx.
(11) Compute Z π/2
0
sin x
cos2x + 3 cos x + 2dx.
In this exercise, you need the change of variable formula:
Z b a
f (x)dx = Z d
c
f (h(t))h0(t)dt,
where x = h(t) is an increasing differentiable function on [c, d] so that h(c) = a and h(d) = b.
(12) Let F (x) = Z x2
0
sin(θ2)dθ for x ∈ R. Compute F0(1).
