# TEXAS A&amp;M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 308-502 ```TEXAS A&amp;M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308-502
Exam 2 version A, 31 Oct 2014
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
In all questions no analytical work =⇒ no points!
1.
Solve the initial value problem
y 00 − y 0 − 6y = e3t ,
using the method of your choice.
y(0) = 1, y 0 (0) = 0,
2.
Find the general solution to
y 00 + 2y 0 + y = t−2 e−t .
3.
A series circuit has a capacitor of C = 0.25 &times; 10−6 F, a resistor of R = 5 &times; 103 Ω, and
an inductor of L = 1H. The initial charge on the capacitor is zero. A 12-volt battery
is connected to the circuit and the circuit is closed at t = 0. What’s the charge on the
capacitor after a long time? Give a quick physics explanation and a full proof using the
solution of the differential equation
LQ00 + RQ0 +
1
Q = E(t).
C
4.
Prove that the Laplace transform of the function

sin(t), 0 &lt; t &lt; π,
g(t) =
0,
t≥π
is equal to
1
1 + e−πs .
+1
You may do it using the definition of Laplace transform or by expressing g(t) in terms of
the Heaviside function u(t) and using the formulas.