MATH 308, Spring 2016
Take Home QUIZ # 5
Due: Thursday 02 / 18 / 2016. In Class.
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.
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1. (10 pts.) Find the solution of
x
.
dy
= 3y − 2x3
dx
2. (10 pts.) A young person with no initial capital invests k dollars per year into equity with average annualized
return 7.5 compounded continuously. Determine k so that 1 million is available for retirement in 40 years
3. (10 pts.) Determine (without solving the problem) an interval in which the solution of the given initial value
problem is certain to exist.
(4 − t2 )y 0 + 2ty = 3t2 ;
y(−3) = 1
.
4. (10 pts.) Verify that y1 (t) = t2 and y2 (t) = t−1 are two solutions of the dierential equation
t2 y 00 − 2y = 0;
t>0
.
Find the Wronskian W (y1 , y2 )(t) and explain if the above two functions form a fundamental set of solutions
5. (20 pts) Another equation that has been used to model population growth is the Gompertz equation
dy
= ryln(K/y)
dt
.
where r and K are positive constants.
a) Find the critical points, and determine whether each is asymptotically stable or unstable
b) For 0 ≤ y ≤ K , determine where the graph of y versus t is concave up and where it is concave down.
a) Solve the equation subject to the initial condition y(0) = y0 .
6. (10 pts ) Find the solution of
(esinx + 2y)y 0 + yesinx cosx − 1 = 0
.
7. (10 pts.) For the equation
y 00 + 4y 0 + (6 + α)y = 0
.
Find the range of α for which the equation has at least one solution which grows unbounded as t → ∞.
8. (10 pts.)
Find the solution of the given initial value problems
a) y 00 − 2y 0 + 5y = 0; y(π/2) = 0, y 0 (π/2) = 2
.
b) y 00 + 4y 0 + 4y = 0; y(−1) = 2, y 0 (−1) = 1
.