1. Solve the differential equation y'' + 2y' + y = 0, subject to the initial conditions y(0) = 1 and y'(0) = -2.

2. 2. A population of rabbits is growing exponentially, such that the rate of change of the population is proportional to the size of the population. If there are initially 100 rabbits and the population doubles every year, find a differential equation that models this situation.

3. 3. A tank initially contains 100 liters of water with 1 kg of salt dissolved in it. Water containing 0.1 kg of salt per liter is entering the tank at a rate of 10 liters per minute, while the well-mixed solution is leaving the tank at the same rate. Write a differential equation for the amount of salt in the tank at any time t, and solve it subject to the initial condition that there is 1 kg of salt in the tank at t = 0.

4. 4. A spring-mass system is modeled by the differential equation y'' + 2y' + 2y = sin(t), where y represents the displacement of the mass from its equilibrium position. Find the general solution to this equation.

5. 5. A tank is being drained by a pump that can remove 10 liters of water per minute. The tank initially contains 100 liters of water. Write a differential equation for the volume of water in the tank at any time t, and solve it subject to the initial condition that there is 100 liters of water in the tank at t = 0. What is the volume of water in the tank after 10 minutes?

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