Differential Equations. ODE. Complete solution. Thank you so much

instructions: Solve the following items. Write your answers and complete solutions on a separate sheet of papeL 1. What type of ODE is obtained in Example 4.1? Justify your answer by expressing the ODE in standard form. 2. Show the complete solution that prove that the particular solution to the ODE is f (t) = 2,000 (1 + Ce't). 3. Find a particular solution to the differential equation given the condition that initially, the pond contains 100 grams of pollutants. (Hint: at t = 0, f(0) = 100.) 4. If t approaches 00, what will be the concentration of pollutants? 1. Given the temperature of the water inside the cooler and initial temperature of the heated liquid, determine the solution to the mathematical model given by equation (1}. 2. Given the terminal time (use minutes as the unit of measure) and the new temperature of the liquid, solve for the constant y. Example 4.1 Water ows into one end of a pond at the rate of 10m3/min and out of the other end at the same rate. The volume of the pond is 10001113. Initially, the pond contains 100g pollutants. The concentration of the pollutants in the water owing into the pond is 29/7113. Upon entering the pond, the pollutants spread so quickly that the concentration of pollutants is uniform throughout the pond. If f (t) is the concentration of pollutants in the pond, find a formula for the rate of change in the concentration of pollutants over time. Solution: The concentration of inowing pollutants is 2g/m3 and fluid-in rate is 10m3/min. Hence, 2g 10m3 , Ratem = () = ZOg/mm m3 min The current concentration of pollutants in the system is given by 2 10:: ogma' Since, uid-out rate is equal to uid-in rate, R t ( f9 ) 10m3 f / . a e = = mm 0'\" 1000m3 min 1009 Combining these equations, we get = 20 _ L