I.
Write TRUE
if the statement is correct and FALSE otherwise. (2 points each)
1.
If a curve
intersects a family of curves at a constant angle, then it is called an
orthogonal trajectory of the given family.
2.
If the
differential equation of a family of curves is
= 3x2tan 2y, then the
differential equation of its orthogonal trajectories is
= -3x2tan 2y.
3.
If the lines
corresponding to the linear coefficients of (a1x + b1y + c1)dx + (a2x + b2y + c2)dy = 0 are parallel, then the differential equation is solved by
substitution.
4.
2ydx + x(x2ln y – 1)dy = 0 is a Bernoulli’s equation.
5.
Kirchhoff’s
current law is used to solve a circuit containing resistance and capacitance
connected in series with an electromotive force.
II.
Obtain a family
of solutions for (5x + 3
)dx + 2x
dy = 0. (15 points)
III.
Solve the equation
(y – 2)dx – (x – y – 1)dy = 0. (15 points)
IV.
Solve the following:
(15 points each)
1.
The population
of a barangay increases at a rate proportional to the number of inhabitants at
time t. If the population has doubled 5 years and tripled in 10 years,
what was the initial population?
2.
A thermometer
is removed from a room where the air temperature is 700F and taken
outside where the temperature is 100F. After 2 minutes, the
thermometer reads 500F. How long will it take for the thermometer to
reach 150F?
3.
A large tank is
partially filled with 100 gallons of water in which 10 lbs of salt are
dissolved. Brine containing
pound of salt per gallon is pumped into the
tank at a rate of 6
. The well-stirred solution is then
pumped out at 4
. Find the amount of salt in the tank
after 20 minutes.
4.
An inductance
of 4 henrys and a resistance of 30 ohms are connected in series with an
impressed voltage of 220 volts. If the initial current is zero, find the current
after 0.01 second.
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