5. A 10kg mass is attached to a spring hanging from a ceiling. This causes the spring to stretch 2m on coming to rest at equilibrium. At time t = 0 an external force f (t) = 20cos4t is applied to the system. Let the damping constant B = 3, formulate a differential equation describing the system and use the method of undetermined coefficients to determine the steady state solution of the system and sketch one cycle of the sine wave form of the steady state.

Find the general solution of the following differential equation: d²x 1 dx dt² + = t²(x) ¹, t dt The given ODE satisfies x(1) = 2 and (1) = 1. Note that x = d. Hints: • Classify the ODE and make preliminary analysis for it. • The given ODE in this question can be solved using the knowledge you have gained so far, especially Topics 3 and 5.

A tank contains 460 gallons of water and 45 oz of salt. Water containing a salt concentration of 1/7(1+1/4sint) oz/gal flows into the tank at a rate of 9 gal/min, and the mixture in the tank flows out at the same rate. The long-time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation? Round the values to two decimal places. Plz correct solution.

A mass weighing 16 lb stretches a spring 1/4 feet. The mass is attached to a viscous damper with a damping constant of 2 lb-s/ft and is set in motion from its equilibrium position with a downward velocity of 1/2 ft/s. a) Set up the initial value problem (differential equation). b) Solve the equation to find the position of the mass y at any time t. c) Determine when the mass first returns to its equilibrium position. Assume that the y-axis is directed downward.