Name: Test 1 F19 Mehrdad Khosravi
1. Find the motion of the mass-spring system with mass 0.125 kg, damping 0, spring constant 1.125 kg/sec2, and driving force cos t − 4 sin t [nt], assuming zero initial displacement and velocity. For what frequency of the driving force would you get resonance?
2. Find the steady state current in the RLC-circuit when R = 4Ω , L = 0.5 H, C = .1 F, and E = 500sin2t V.
3. A cantilever beam of length L is embedded at its right end and a horizontal tensile force of P pounds is applied to its free left end. The deflection y(x) of the beam can be shown to satisfy the differential equation
EIy′′ =Py−w(x)x. 2
Find the deflection of the cantilever beam if w(x) = w0x, 0 < x < L, and y(0) = 0, y′(L) = 0, where the origin is placed at the free end.
4. Two tanks are filled with 200 gallons of water each. There is 100 lbs of salt dissolved in the first tank and 200 lbs of salt in the second tank. We have pure water entering the first tank at a rate of 12 gal/min and the well mixed solution exits into the second tank at a rate of 16 gal/min. The well mixed solution in the second tank back flows in the first tank at a rate of 4 gal/min and exist into a drain at a rate of 12 gal/min. Set up and solve a system of differential equations for the amount of salts y1(t) in tank 1 and y2(t) in tank 2.
5. Find the limit state of the Markov process modeled by the given matrix.
0.6 0.1 0.2 0.4 0.1 0.4
0 0.8 0.4
6. A model for the spread of contagious diseases is obtained by assuming that the rate of spread is proportional to the number of contacts between infected and noninflected persons, who are assumed to move freely among each other. Set up the model. Find the equilibrium solution and indicate their stability or instability. Solve the ODE. Find the limit of the proportion of infected persons as t → ∞ and explain what it means.
7. Consider a LRC-circuit with the model 1
C I1dt+R(I1 −I2)=0 LI2′ + R(I2 − I1) = 0
Find a general solution, assuming that R = 3Ω, L = 44 H, C = 1/12 F. 8. Use the Laplace transform to solve:
dx −4x+d3y =6sint dt dt3
dx +2x−2d3y =0 dt dt3
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