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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 (-∞,∞).
- {x2, x+1, x-3} ans, W=8, linearly independent
- {3e2x, e2x} ans, W=0, linearly dependent
- {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 (-∞,∞).
- {ln x, ln x2} ans, W=0, linearly dependent
- {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?
If a string of length l is initially at rest in equilibrium position and each of its points is given the velocity dy/dt= b sin^3πx/l find the displacement
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.
the temperature of both end of a uniform metal bar is maintained at 0⁰C. its length is 10 units. The temperature of the bar is modeled by the equation, dT(x,t)/dt = 4 d²T(x,t)/dx². Determine T(x,t) given that at t = 0 the temperature dependence of the bar is T(x,0) = x(10-x).
Initial 100 milligram of a radio active substance was present.After 6 hour the mass has been decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time t,find the amount remaining after 24hours.
y'''+4y'=ex cos2x write the given differential equation in the form pf L(y)=g(x),where L is a linear differential operator with constant .IF possible , factor L.
#339914 on Dec 2023