Differential Equations Answers

y''-25y=0; y1=e^5x the indicated function y1(x) is a solution of the given differential eqution.Use reduction of order or formula as instructed, to find a second solution y2(x).

A metal bar with an initial temperature, 𝑇0, in the interval of 30°C ≤ 𝑇0 ≤ 35°C is dropped into a container of boiling water (100°C). The temperature of the metal bar, 𝑇 at any time, 𝑡 satisfies the following Newton’s Law of Cooling model 𝑑𝑇 𝑑𝑡 = −𝑘(𝑇 − 𝑇𝑚) where 𝑇𝑚 is the ambient temperature and 𝑘 is the constant. After 5 seconds, the temperature of the bar, 𝑇1 is in the interval of 40°C ≤ 𝑇1 ≤ 50°C. a. Find the equation that models the temperature of the metal bar, 𝑇 at any time, 𝑡 (choose a value of 𝑇0 and 𝑇1 from the given intervals, respectively). By using an appropriate analytical method, solve the derived model and explain the reason for the selection of the method. b. Compute the temperature of the metal bar after 100 seconds by using the derived model in Part 1(a) with THREE (3) different numerical methods with step size, ℎ = 10 seconds. Select the best numerical method to compute the temperature of the metal bar and justify your answer

A certain radioactive material is decaying at a rate proportional to the amount present.

If a sample of 50 grams of the material was present initially and after 2 hours the sample lost 10% of its mass, find:

(a) An expression for the mass of the material remaining at any time t.

(b) The mass of the material after 4 hours.

(c) The time at which the material has decayed to one half of its initial mass.

Find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on (-∞,∞).

1. {x², x+1, x-3}                ans, W=8, linearly independent
2. {3e^2x, e^2x}                     ans, W=0, linearly dependent
3. {x², x³, x⁴}                    ans, W=2x^6, linearly independent

Find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on (-∞,∞).

1. {ln x, ln x²}                   ans, W=0, linearly dependent
2. {2+x, 1-x, 3+x²}           ans, W=-6, linearly independent 

A 50 gallons tank initially contains 10 gal of fresh water. At t = 0, a brine solution

containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gal/min, while the

well-stirred mixture leaves the tank at the rate of 2 gal/min. Find

a. the amount of time required for overflow to occur

b. the amount of salt in the tank at the moment of overflow

A metal bar at a temperature of 100 deg. fahrenheit is placed in a room at a constant temperature of 0 deg. fahrenheit. If after 20 minutes the temperature of the bar is 50 deg. Fahrenheit, find: a) the time it will take the bar to reach a temperature of 25 deg. Fahrenheit? b) the temperature of the bar after 10 minutes?

reduce the equation

∇²ψ + [k² + f(ρ) + (1/ρ²)g(φ) + h(z)]ψ = 0

to a set of ODEs by the method of separation of variables.