MATH 308, Spring 2016
Take Home QUIZ # 10
Due: Thursday 04 / 15 / 2016
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SECTION #:
INSTRUCTOR: Dr. Marco A. Roque Sol
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1. (10 pts.) Determine φ00 (x0 ), φ000 (x0 ), andφ(4) (x0 ) for the given point x0 if y = φ(x)is a solution of the given initial
value problem
y 00 + xy 0 + y = 0;
y(0) = 1, y 0 (0) = 0
2. (10 pts.) Determine a lower bound for the radius of convergence of series solutions about each given point x0
for the given dierential equation.
(1 + x3 )y 00 + 4xy 0 + 6xy = 0;
x0 = 0.
x0 = 4
3. (10 pts.) The Chebyshev Equation. The Chebyshev dierential equation is
(1 − x2 )y 00 − xy 0 + α2 y = 0,
−∞ < x < ∞
where α is a constant.
(a) Determine two solutions in powers of x for |x| < 1, and show that they form a fundamental set of solutions.
(b) Show that if α is a nonnegative integer n, then there is a polynomial solution of degree n. These polynomials,
when properly normalized, are called the Chebyshev polynomials. They are very useful in problems that require
a polynomial approximation to a function dened on −1 ≤ x ≤ 1.
(c) Find a polynomial solution for each of the cases α = n = 0, 1, 2, 3.
4. (10 pts.) The Legendre Equation. The Legendre equation is
(1 − x2 )y 00 − 2xy 0 + α(α + 1)y = 0, x0 = 0
(a) Determine two solutions in powers of x for |x| < 1, and show that they form a fundamental set of solutions.
(b) Show that if α is zero or a positive even integer 2n, the series solution y1 reduces to a polynomial of degree
2n containing only even powers of x. Find the polynomials corresponding to α = 0, 2, 4. Show that if α is a
positive odd integer 2n + 1, the series solution y2 reduces to a polynomial of degree 2n + 1 containing only odd
powers of x. Find the polynomials corresponding to α = 1, 3, and5 .
(c) The Legendre polynomial Pn (x) is dened as the polynomial solution of the Legendre equation with α = n
that also satises the condition Pn (1) = 1. Find the Legendre polynomials P0 (x), ..., P5 (x). Plot the graphs of
P0 (x), ..., P5 (x) fo r − 1 ≤ x ≤ 1.
5. (10 pts.) Find all singular points of the given equation and determine whether each one is regular or irregular
(a)
xy 00 + (1 − x)y 0 + xy = 0
(b)
(c)
x2 (1 − x)y 00 + (x − 2)y 0 − 3xy = 0
(1 − x2 )y 00 + x(1 − x)y 0 + (1 + x)y = 0
6. (10 pts.) Determine the general solution of the given dierential equation that is valid in any interval not
including the singular point.
(a)
x2 y 00 + 4xy 0 + 2y = 0
(b)
x2 y 00 − 3xy 0 + 4y = 0
(c)
x2 y 00 − xy 0 + y = 0
7. (30 pts.)
(a) In the following Problem:
2x2 y 00 + 3xy 0 + (2x2 − 1)y = 0
Show that the given dierential equation has a regular singular point at x = 0. Determine the indicial equation,
the recurrence relation, and the roots of the indicial equation. Find the series solution (x > 0) corresponding to
the larger root. If the roots are unequal and do not dier by an integer, nd the series solution corresponding
to the smaller root also.
(b) The Bessel equation of order zero is
x2 y 00 + xy 0 + x2 y = 0
(bi) Show that x = 0 is a regular singular point.
(bii) Show that the roots of the indicial equation are r1 = r2 = 0.
(biii) Show that one solution for x > 0 is
.
∞
J0 (x) = 1 +
X (−1)n x2n
22n (n!)2
n=1
(biv) Show that the series for J0 (x) converges for all x. The function J0 is known as the Bessel function of the
rst kind of order zero.
(c) The Bessel Equation of Order One-Half is
1
x2 y 00 + xy 0 + (x2 − )y = 0
4
Show that x = 0 is a regular singular point and that the roots of the indicial equation are r1 = 1/2, r2 = −1/2.
Find the two solutions J1/2 and J−1/2 .
8. (10 pts.)
(a) The Gamma Function. The gamma function is denoted by Γ(p) and is dened by the
Z
Γ(p + 1) =
∞
e−x xp dt
0
The integral converges as x → ∞ for all p. For p < 0 it is also improper at x = 0, because the integrand becomes
unbounded as x → 0. However, the integral can be shown to converge at x = 0 for p > −1.
(a) Show that, for p > 0,
Γ(p + 1) = pΓ(p)
(b) Show that Γ(1) = 1.
(c) If p is a positive integer n, show that
Γ(n + 1) = n!
.
Since Γ(p) is also dened when p is not an integer, this function provides an extension of the factorial function
to nonintegral values of the independent variable. Note that it is also consistent to dene 0! = 1.
(d) Show that, for p > 0,
Γ(p + n)
Γ(p)
p(p + 1)(p + 2) . . . (p + n − 1) =
.
Thus Γ(p) can be determined for all positive values
of p if Γ(p) is known in a single interval of unit length - say,
√
0 < p ≤ 1. It is possible to show that Γ( 21 ) = π , Find Γ( 23 ) and Γ( 11
2 )
9. (20 pts.) Consider the Laplace transform of tp , where p > −1
(a) Referring to the previous Problem, show that
L {tp } =
.
∞
Z
sp+1
0
=
Γ(p + 1)
.
sp+1
Z
1
e−st tp dt =
∞
e−s xp
0
s>0
.
(b) Let p be a positive integer n in part (a); show that
L {tn } =
n!
,
sn+1
s>0
(c) Show that
2
L {t−1‘/2 } = √
s
Z
∞
2
e−x dx
0
It is possible to show that
Z
√
∞
2
e−x dx =
0
( using double integrationg with polar coordinates;
R ∞ R ∞ r2
e rdrdθ = π4 ). Hence
0
0
L {t
π
2
2
∞ −x2
e
dx
0
R
−1‘/2
r
=
R
π
,
s
s>0
π
,
(2s3/2
s>0
}=
R
∞ −x2
e
dx
0
∞ −y 2
e dy
0
=
R∞R∞
0
0
e−x
2
+y 2
(d) Show that
√
L {t
1‘/2
}=
10. (20 pts.) In each of Problems, use the Laplace transform to solve the given initial value problem.In each of
Problems 11 through 23, use the Laplace transform to solve the given initial value problem
(a)
(b)
y 00 + ω 2 y = cos(2t);
ω 2 6= 4,
y(0) = 1,
y 0 (0) = 0
y 00 + 2y 0 + y = 4e−t ;
ω 2 6= 4,
y(0) = 2,
y 0 (0) = 1
dxdy =