Consider the initial-value problem

1. (12 points) Consider the initial-value problem What is the largest interval for which this initial-value problem has a solution? Explain your reasoning. Hint: Do NOT try to solve the initial-value problem. 2. Solve the following equations and initial-value problems. (a) (15 points) (1 + yQJy' = y 0052:- You may leave your answer dened implicitly. (b) (18 points) 932+y1+($4y)y': 9(1J=4 (c) (17 points) 2:; r_:32 y+I 19$ 3. A tank starts with 100 gallons of a salt-water solution with 50 grams of salt. Salt water with a concen- tration of l +loost 4 2 grams per gallon ows into the tank at a rate of 2 gallons per minute (t is measured in minutes). The incoming salt water is instantly mixed with the salt water in the tank and the salt water in the tank ows out at a rate of 2 gallons per minute. (a) (15 points) Give an initial-value problem describing the evolution of the amount of salt AB) in the tank at time t. (b) (10 points) Suppose the entering solution remains the same, but the outow is 3 gallons per minute. Ebcplain how the differential equation in your initial-value problem would change. Hint: The volume of the water in the tank is now 100 t. 4. (18 points) Solve the equation my" + y" = 4:13. Hint: Find and solve an equation for z = y', then use that solution to nd 3