Find the equation of the family of orthogonal trajectories of the system of parabolas \(y^2=2x+c\).
A\(y=Ce^{-x}\)
B\(y=Ce^{2x}\)
C\(y=Ce^x\)
D\(y=Ce^{-2x}\)
Mathematics · Question Bank
Differential Equations
39 of 42 questions match · page 2 of 2. Answer in your head first, then check the solution.
According to Newton's law of cooling, the rate at which a substance cools in air is directly proportional to the difference between the temperature of the substance and that of air. If the temperature of the air is \(30^\circ\) and the substance cools from \(100^\circ\) to \(70^\circ\) in 15 minutes, how long will it take to cool \(100^\circ\) to \(50^\circ\)?
A33.59 min.
B43.60 min
C35.39 min
D45.30 min
An object falls from rest in a medium offering a resistance. The velocity of the object before the object reaches the ground is given by the differential equation \(dV/dt+V/10=32\), ft/sec. What is the velocity of the object one second after it falls?
A40.54
B38.65
C30.45
D34.12
In a tank are 100 liters of brine containing 50 kg. total of dissolved salt. Pure water is allowed to run into the tank at the rate of 3 liters a minute. Brine runs out of the tank at the rate of 2 liters a minute. The instantaneous concentration in the tank is kept uniform by stirring. How much salt is in the tank at the end of one hour?
A15.45 kg.
B19.53 kg.
C12.62 kg.
D20.62 kg.
A tank initially holds 100 gallons of salt solution in which 50 lbs of salt has been dissolved. A pipe fills the tank with brine at the rate of 3 gpm, containing 2 lbs of dissolved salt per gallon. Assuming that the mixture is kept uniform by stirring, a drain pipe draws out of the tank the mixture at 2 gpm. Find the amount of salt in the tank at the end of 30 minutes.
A171.24 lbs.
B124.11 lbs.
C143.25 lbs.
D105.12 lbs.
If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuously compounded account?
AP5,750
BP6,750
CP7,500
DP6,350
A nominal interest of 3% compounded continuously is given on the account. What is the accumulated amount of P10,000 after 10 years?
AP13,620.10
BP13,500.10
CP13,650.20
DP13,498.60
The equation \(y^2 = cx\) is the general solution of:
A\(y' = \dfrac{2y}{x}\)
B\(y' = \dfrac{2x}{y}\)
C\(y' = \dfrac{y}{2x}\)
D\(y' = \dfrac{x}{2y}\)
Find the general solution of \((D^2 - D + 2)y = 0\).
A\(y = e^{x/2}\left(C_1\sin\tfrac{\sqrt{7}}{2}x + C_2\cos\tfrac{\sqrt{7}}{2}x\right)\)
B\(y = e^{x/2}\left(C_1\sin\tfrac{\sqrt{7}}{2}x - C_2\cos\tfrac{\sqrt{7}}{2}x\right)\)
C\(y = e^{x/2}\left(C_1\cos\tfrac{\sqrt{7}}{2}x + C_2\sin\tfrac{\sqrt{7}}{2}x\right)\)
D\(y = e^{x/2}\left(C_1\cos\tfrac{\sqrt{7}}{2}x - C_2\sin\tfrac{\sqrt{7}}{2}x\right)\)
A steel ball at 110°C cools in 8 min to 90°C in a room at 30°C. Find the temperature of the ball after 20 minutes.
A58.97°C
B68.97°C
C78.97°C
D88.97°C
A certain radioactive substance has a half-life of 3 years. If 10 grams are present initially, how much of the substance remains after 9 years?
A1.50 grams
B1.25 grams
C2.50 grams
D1.75 grams
Find A for which \(y = Ae^x\) will satisfy \(y'' - 2y' = 4e^x\).
A−1
B−2
C−3
D−4
What is the general solution of \((D^4 - 1)y(t) = 0\)?
A\(y = C_1e^t + C_2e^{-t} + C_3\cos t + C_4\sin t\)
B\(y = C_1e^t + C_2e^{-t} + C_3te^t + C_4te^{-t}\)
C\(y = C_1e^t + C_2e^{-t}\)
D\(y = C_1e^t + C_2te^{-t}\)
Find the general solution of \(y'' + 10y' + 41y = 0\).
A\(y = e^{-5x}(C_1\cos 4x + C_2\sin 4x)\)
B\(y = e^{5x}(C_1\cos 4x + C_2\sin 4x)\)
C\(y = e^{-4x}(C_1\cos 5x + C_2\sin 5x)\)
D\(y = e^{4x}(C_1\cos 5x + C_2\sin 5x)\)
Find the general solution of \(y' + \dfrac{x}{y} = 0\).
A\(x^2 + 2y^2 = C\)
B\(x^2 + y^2 = C\)
C\(x^2 - 2y^2 = C\)
D\(x^2 - y^2 = C\)
The equation \(y^2 = cx\) is the general solution of
A\(y' = \dfrac{2y}{x}\)
B\(y' = \dfrac{2x}{y}\)
C\(y' = \dfrac{y}{2x}\)
D\(y' = \dfrac{x}{2y}\)
Solve the differential equation \(y' = \dfrac{y}{2x}\).
Ay = cx
By² = cx
Cy = cx²
Dy³ = cx
A laboratory has a 75-gram sample of radioactive material. The half-life of the material is 10 days. What is the mass of the laboratory’s sample remaining after 30 days?
A9.375 grams
B11.25 grams
C12.5 grams
D22.5 grams
A motorboat weighs 32,000 lb and its motor provides a thrust of 5,000 lb. Assume that the water resistance is 100 pounds for each foot per second of the speed v of the boat. Then \(1{,}000\dfrac{dv}{dt} = 5{,}000 - 100v\). If the boat starts from rest, what is the maximum velocity that it can attain?
A20 ft/s
B25 ft/s
C40 ft/s
D50 ft/s