TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308
Exam 1 version A, 28 Sep 2012
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
No detailed analytical work — no points.
1.
Find the solution of
ty 0 + 2y = sin t,
y(π/2) = 0.
2.
Find the general solution to
x2 e−y dx + y dy = 0.
3.
Find the general solution to
(x + tan−1 y)dx +
x+y
dy = 0.
1 + y2
4.
A 400-gal tank initially contains 100-gal of pure water. Brine containing 1 lb/gal of salt
enters the tank at the rate of 5 gal/s and the well-mixed brine in the tank flows out at the
rate of 4 gal/s. Write the differential equation governing the amount of salt in the tank.
Solve it. How much salt will the tank contain when it is full of brine?
5.
Find the solution y(t) of the initial value problem (IVP) for all values of a
y 00 + 4y 0 + (4 − a2 )y = 0,
y(0) = 0, y 0 (0) = 1.
(Bonus question: 2pnts) For each value of t calculate the limit of the solution as a → 0.
Does it make sense?
Points:
/25