Problem 3 continued... :2) (4 points) Evaluate the indefinite integral fdl. 80 d) (8 points) Use your work from part (c) to solve the initial value problem Explicitlyr solve for N as a function oft in your final answer. Note: This will involve some algebra. Be sure to show all your steps. e) (2 points) Use your solution to the IVP from Part c to determine tlim N(t). Does this agree with your answer in Part Ace a? 3) A logistic growth model can be used model population growth when there are limited resources to support the population. For instance, the model can be used to track the population growth of a species of fish in a pond, where the limited space and availability of food will place constraints on how the fish population can increase in time. The differential equation that models logistic growth is dN N E=kN(1Elv where N03) is the population at time t. k is the growth rate, and L is a constant. For the purpose of this problem. we will take L = 20 and k = 1/4. a) (5 points) The direction field for the differential equation is shown below. f/f/ ////////////////////////////// //// /////////////2//////////////// 11/; 2///////////////////////////// /// ////////////////////////////// /// ////z///////////////////////// //// ////////////////////////////// //// ////////////////////////////// //// ////////////////////////////// 111/ /////IfIll/!////////////////// ///w ////////////////////////////// /// ////////////////////////////// //// ////////////////////////////// //// ////////////////////////////// //// //////////////////1/1/1111;11/ //// 11//////////////////////////// ///A xxx/xxx;////////////////////// //// ////////////////////////////// //// 2/;/////////////////////////, J2}! ////////////////////////////, xx}; z/z/fzx/zfzlezzzfl'JJ/IIIII, dN N i) Sketch the solution to the initial value problem 3 UV (1 ) , N(Cl) = 2 for t Z 0. ii) From the direction field, what do you expect the maximum population to be? That is, make a conjecture about tlim NH) based on the direction field. Write your answer in the box. 4w 11m N6) = troo 1 b) (3 points) Substitute the values I: = a and L = 20 into the equation and show that the differential equation can be brought into the form 80 (EN = dt. N(2O a N)