MATH 308, Spring 2016
Take Home QUIZ # 9
Due: Thursday 04 / 07 / 2016
Print name (LAST, First):
SECTION #:
INSTRUCTOR: Dr. Marco A. Roque Sol
THE AGGIE CODE OF HONOR
"An Aggie does not lie, cheat, or steal, or tolerate those who do." By signing below, you indicate that all
work is your own and that you have neither given nor received help from any external sources.
SIGNATURE:
1. (10 pts.) Find the solution of the given initial value problem.
y 000 − y 00 + y 0 − y = sect;
y(0) = 2, y 0 (0) = −1, y 00 (0) = 1
Then, using MATLAB, plot a graph of the solution.
2. (10 pts.) Determine the general solution of the given dierential equation.
y 000 + y 00 + y 0 + y = 4t + et
3. (10 pts.) Find the solution of the given initial value problem,
y (4) − 4y 000 + 4y 00 = 0;
y(1) = −1, y 0 (1) = 2, y 00 (1) = 0, y 000 (1) = 0
and plot, using MATLAB, its graph. How does the solution behave as t → ∞?
4. (10 pts.) The Hermite Equation. The equation
y 00 − 2xy 0 + λy = 0,
−∞ < x < ∞
where λ is a constant, is known as the Hermite equation. It is an important equation in mathematical physics.
(a) Find the rst four terms in each of two solutions about x = 0 and show that they form a fundamental set of
solutions.
(b) Observe that if λ is a nonnegative even integer, then one or the other of the series solutions terminates and
becomes a polynomial. Find the polynomial solutions for λ = 0, 2, 4, 6, 8, and 10. Note that each polynomial is
determined only up to a multiplicative constant.
(c) The Hermite polynomial Hn (x) is dened as the polynomial solution of the Hermite equation with λ = 2n
for which the coecient of xn is 2n. Find H0 (x), ..., H5 (x).
5. (10 pts.) For the following ODE
y 00 − xy 0 − y = 0, x0 = 0
(a) Seek power series solutions of the given dierential equation about the given point x0 ; nd the recurrence
relation.
(b) Find the rst 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, nd the general term in each solution.