Differential Equations Answers

Solve by the power series method d²y/dx² -y =0

A tightly stretched string of length"l"has it's ends fastened at x=0andx=l. The midpoint of the string is taken the height h and released. find the initial displacement function at timet=0?

Find inverse laplace transform

(2s-56)/(s^2-4s-12)

Find fourier transform of

f(t)={ e^2it, -1<t<1

0,otherwise

Solve the initial/moving-boundary problem

u_xx-u_tt = 0 , 0 < x < infinity, 0 < t < 2x

u(x,0) =u₀(x) , 0 <= x < infinity

u_t(x,0)=v₀(x) , 0 <= x < infinity

u(x,2x)=h(x) , x >= 0

where u₀(x), v₀(x), h(x) are twice continuously differentiable on the domain

1. A certain planet has an average surface area that is 2.5% larger than the earth and a gravitational acceleration that is one fifth times than that of the earth’s gravitational acceleration. What is the velocity of escape on that planet? Ans. 5.04 km/s
2. An object weighing 16 pounds is dropped from the top of a 1000 ft building with an initial velocity of 10 ft/sec. as it falls, the object encounters air resistance that is equal to 1/4 v. What is the velocity of the object at the end of 5 seconds? What is the limiting velocity? Ans. 59.57 ft/s; 64 ft/s

1. The radius of the planet mars is approximately 3,396 km and has a gravitational acceleration equal to three eighths times than that of the earth. What is the velocity of escape? Ans. 5 km/s
2. Initially a tank holds 100 gal of a brine solution containing 20lb of salt. At t=0, fresh water is poured into the tank at the rate of 5 gal/min, while stirred mixture leaves the tank at the rate 3 gal/min. what is the amount of salt in the tank after 1 hours? How long will it take for the take to contain 10 lb of salt? Ans. 6.12 lb; 29.37 min.

1.   A tank initially contains 50 gal of fresh water. At t=0, brine solution containing 2 lb of salt per gallon is poured into the tank at the rate of 12 gal/min, while the well stirred mixture leaves the tank at the rate of 8 gal/min. what is the amount of salt at the end of 5 minutes? How long will it take to obtain an amount of 50 lb? ans. 88.98 lb; 2.47 min
2. Given an RC series circuit that has an emf source of 50 volts, a resistance of 20 kilo ohms, a capacitance of 6 microfarad and the initial charge of the capacitor of the capacitor is 1 micro coulomb. What is the charge in the capacitor at the end of 0.01 second? What is the current in the circuit at the end of 0.05 seconds? Ans. 24.91µC; 1.64mA

1. An object has a mass of 2kg is dropped from the top of a building 30 meters tall. the initial velocity is zero. As it falls, the object encounters air resistance that is equal to 1/3 v. what is the velocity and attitude of the object after 1.5 seconds? Ans. 13.02m/s; 19.83m
2. A tank initially contains 50 gal of fresh water. At t=0, brine solution containing 2 lb of salt per gallon is poured into the tank at the rate of 12 gal/min, while the well stirred mixture leaves the tank at the rate of 8 gal/min. what is the amount of salt at the end of 5 minutes? How long will it take to obtain an amount of 50 lb? ans. 88098 lb; 2.47 min