MATH24 - Long Quiz #2 - 2nd TERM - 20112012

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.