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(a) Show that an members of the family y-ve-1ฐ are-lutkn-ed Medaterntial mpalii (b) Use part (a) to find a fornvula for the solution to the ini- tial value problem v (0)-2. Then, sketch your solution on the slope field shown to the right. 2. Figure 1. Slope field for 3. (a) Show that all members of the family yarlutions of the disferential equation (b) Find the solution to the initial value problem ry'-tra-Zy·y(1)-S. 4. For what nonzero values of k does the function y e solve the differential equation " --6y 07 . A popalation is modeled by the dilferential oquation 12P 12 (a) For what values of P is the population increasing? (b) For what values of P is the population decreasing? 6. Below, you are given 8 differential equations. Match each differential equation with the appropriate direction field (given on page 5). Then, complete parts (a)-(c) below (1) y' = 1 (IV) ' cos(r) (VIII (VII) 0 (a) For each of the direction fields, draw in the solution that satisfies the initial condition v(0) 1 (b) Can you guess a formula for any of the solutions you drew in for part (a)? Check to see if you 're right! Definition. A constant solution to a differential equation is called an equalibriun solution. Graphically an equilibrium solution is a horizontal line, that is, a solution of the form y c for some constant c. 7. Use the above definition of equilibrium solution to answer the following (a) Does the differential equation from the Preliminary Example a few pages ago have any equilbrium solutions? If so, what are they? (b) Does the differential equation from Example 2 (preceding the exercises) have any equilibrium solutions? If so, what are they? (c) Does the differential equation from Exercise 5 above have any equilibrium solutions? If so, what are they (d) Which of the differential equations from Exercise 6 above have an equilibrium solution? For those that do, what are the equilibrium solutions? Developed by Jerry Mor (13) 3 -2-101 2 3 -3 -2 10 123 -3 -2 -212 3 LA -3 -2 10 23 -3 -2- oI 2 3 to accompany Calculus, Hughes-Hallett et al, Wiley, 2017

Answer 1

Ang 3a) puHry this into the eGUatio ny'ニ5x3-2y 3 is Hence (j- x3+ 8oth sids are the famly oSolutions s , pluahs info y= χ3+ 、〉 The Solution to the vp Kt kekt and y"- k'ekt ina the se into the equation kt Kt K- Hence k-6 =0 Soluing this we get k 2 and3 The values of k 2 and 3 54) p is increasing when deat>

i-c when (l-P_ 42o 0 mter val open values-of....-P--(。y420。) yhe ne Values

b) P is decreasing when站く0 le when (1-LLO ina when ar < o dt 4200 P1200 when Pis dec reasing