MATH 308, Spring 2017
Take Home QUIZ # 8B
Due: Friday 04/ 21 / 2017
SECTION #:
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INSTRUCTOR: Dr. Marco A. Roque Sol
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work is your own and that you have neither given nor received help from any external sources.
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Directions:Show all your work neatly and concisely. You will be graded not merely on the final answer, but
also on the quality and correctness of the work leading up to it.
1. (20 pts.) In the following Problems:
(a) Seek power series solutions of the given differential equation about the given point x0 ; find the recurrence
relation.
(b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner).
(c) By evaluating the Wronskian W (y1, y2)(x0 ), show that y1 and y2 form a fundamental set of solutions.
(d) If possible, find the general term in each solution.
1) y 00 + k 2 x2 y = 0,
x0 = 0;
k = constant
( Hint: Let n = 4m in the recurrence relation, m = 1, 2, 3, ... )
2) y 00 + xy 0 + 2y = 0,
x0 = 0
2. (30 pts.) In each of Problems shown below, determine a lower bound for the radius of convergence of series
solutions about each given point x 0 for the given differential equation.
1)
(x2 − 2x − 3)y 00 + xy 0 + 4y = 0,
2) xy 00 + y = 0,
x0 = 4;
x0 = −4;
x0 = 0
x0 = 1;
3. (60 pts.) In each of Problems shown below, determine the general solution of the given differential equation that
is valid in any interval not including the singular point.
1) x2 y 00 + 4xy 0 + 2y = 0
2)
(x + 1)2 y 00 + 3(x + 1)y 0 + (3/4)y = 0
3) x2 y 00 − xy 0 + y = 0
4)
(x − 1)2 y 00 + 8(x − 1)y 0 + 12y = 0
5) x2 y 00 − 5xy 0 + 9y = 0
6)
(x − 2)2 y 00 + 5(x − 2)y 0 + 8y = 0
4. (60 pts.) In each of Problems shown below, find all singular points of the given equation and determine whether
each one is regular or irregular.
1) xy 00 + ex y 0 + (3cosx)y = 0
2) y 00 + (ln|x|)y 0 + 3xy = 0
3) x2 y 00 + 2(ex − 1)y 0 + (e−x cosx)y = 0
4) x2 y 00 − 3(sinx)y 0 + (1 + x2 )y = 0
5) xy 00 + y 0 + (cotx)y = 0
6)
(sinx)y 00 + xy 0 + 4y = 0
5. (40 pts.)
a) Find all values of α for which all solutions of x2 y 00 + αxy 0 + (5/2)y = 0 approach zero as x → 0 .
b) Find all values of β for which all solutions of x2 y 00 + βy = 0 approach zero as x → 0 .
c) Find γ so that the solution of the initial value problem x2 y 00 − 2y = 0, y(1) = 1, y 0 (1) = γ is bounded as x → 0.
d) Find all values of α for which all solutions of x2 y 00 + αxy 0 + (5/2)y = 0 approach zero as x → ∞ .
6. (20 pts.)Using the method of reduction of order, show that if r1 is a repeated root of
r(r − 1) + αr + β = 0,
then y1 = xr1 and y2 = xr1 lnx, are solutions of x2 y 00 + αxy 0 + βy = 0 for x > 0.
7. (40 pts.) In each of Problems shown below:
(a) Show that the given differential equation has a regular singular point at x = 0.
(b) Determine the indicial equation, the recurrence relation, and the roots of the indicial equation.
(c) Find the series solution (x > 0) corresponding to the larger root.
(d) If the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller
root also.
1) xy 00 + y = 0
2)
3x2 y 00 + 2xy 0 + x2 y = 0
8. (20 pts.) The Laguerre differential equation is
xy 00 + (1 − x)y 0 + λy = 0.
(a) Show that x = 0 is a regular singular point.
(b) Determine the indicial equation, its roots, and the recurrence relation.
(c) Find one solution (x > 0). Show that if lambda = m, a positive integer, this solution reduces to a polynomial.
When properly normalized, this polynomial is known as the Laguerre polynomial, Lm (x).
9. (20 pts.) In each of Problems shown below:
(a) Find all the regular singular points of the given differential equation.
(b) Determine the indicial equation and the exponents at the singularity for each regular singular point.
1) x2 y 00 + 3(sinx)y 0 − 2y = 0
2)
2x(x + 2)y 00 + y 0 − xy = 0
10. (30 pts.)In each of Problems shown below :
(a) Show that x = 0 is a regular singular point of the given differential equation.
(b) Find the exponents at the singular point x = 0.
(c) Find the first three nonzero terms in each of two solutions (not multiples of each other) about x = 0.
1) xy 00 + y 0 − y = 0
2) xy 00 + 2xy 0 + 6ex y = 0