MAT 21B Lecture 21: MAT 21B – Lecture 21 – Practice Midterm Review

Step 3 Multiply both sides of the differential equation by t and get [ty] = dt Step 4 Integrating both sides and solving for y, we obtain C1C- 2 Int C To satisfy the initial condition, we must have 21n(1)+C-C [In(1)-이. Hence, the solution of the initial-value problem is 2Int 1 Figure 3 Solution of the initial-value problem fy + ty 2, y(1)= 1, t > 0 t > 0 Among all the solution curves that are shown in Fig. 3, th problem is the curve that goes through the point (1, 1) e solution of the > Now Try Exercise In the following section, we present several interesting applications of fre linear differential equations. Check Your Understanding 10.3 1. Using an integrating factor, solve yty 1e 2. Find an integrating factor for the differential equi EXERCISES 10.3 in Exercises 1-6, find an integrating factor for each equation Taket0. In Exercises 21-28, solve the initial-value problem. 22. ty, + y = In t, y(e) = 0, t > 0 Int-1 23. y+ 24, y, 2(10-y), y(0) = 1 10-9e-2t 10+10f In Exercises 7-20, solve the given equation using an integrating factor. Taket>0. 26. ty'-y=-1, y(1) = 1, t>01 27, y, + 2y cos(2t)-2 cos(2t), y(5) = 0 y 28, ty' + y-sin t, y()-0, t > 0-1 cost Technology Exercises 29. Consider the initial-value -1- - 9. 10· y' = 2(20-y) 20 + C'e-2t 12. e +1 11、 y' = .5(35-y) 13. y, + 10+1-0y=府14.3-h-e2t tr" + Cee 15. (1 + tly, + y--I 17. 6y'+ty t 19,y, + y = 2-et y'=-th+10, (a) Is the solution increasing or decreasing whie y(0)=50. 16, y, = e-"(y + 1) _1+ 18. ety' + y = 1 1 + Ce", C'e-e-t 20 , + y = 1 [Hint: Compute y'(0),1 i + Ce-i(t+1)3/2 (b) Find the solution and plot it for OStS t k indicates answers that are in the back of the book. 19. y = 2-ne, + Ce-r 29. (a) y'(0)--40, y(t) is decreasing at t = 0

HW8: Problem 10 Previous Problem List | Next (1 point) We consider the non-homogeneous problem y"-4y, + 4y = 16x2 First we consider the homogeneous problem y"-4y + 4y = 0 : 1) the auxiliary equation is ar2 + br + c = 2) The roots of the auxiliary equation are 3) A fundamental set of solutions is complementary solution ye = ciyi + c2y2 for arbitrary constants q and c2 (enter answers as a comma separated list). (enter answers as a comma separated list). Using these we obtain the the problem y"-4y, + 4y = 16x2 using the rmethod of Next we seek a particular solution y, of the non-homogeneous undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find y, We then find the general solution as a sum of the complementary solution y cyi +022 and a particular solution: y=ye +y,"Finally you are asked to use the general solution to solve an IVP 5) Given the initial conditions y(0) = 8 and y(0) 14 find the unique solution to the IVP 26