Exam 2 Study Guide: MATH 2080: Summer I 2016

Transcription

1 Exam Study Guide: MATH 080: Summer I 016 Dr. Peterson June First Order Problems Solve the following IVP s by inspection (i.e. guessing). Sketch a careful graph of each solution. (a) u u; u(0) 0. (b) u 3u; u(0) 10.. Solve the following IVPs by the integrating factor method showing me all the details of your arguments. (a) u 3u + 4; u(0) 5. (b) u 7u 8; u(0) 30. Do the following Partial Fraction Decomposition integrals. This is at the heart of the solution to the logistics model so they are good exercises. (a) w/(w 5)(w 8)dw. (b) /(x + 10)(3x 40)dx. 4. Sketch the solution to the following logistic problems without doing the actual solution via our long integration technique. (a) u.06u(400 u); u(0) 450. (b) u 13u(1300 u); u(0) Solve the following logistic problems the long way. Make sure you show all the steps do all the simplifications and explain how you handle the absolute value in the logarithm terms. (a) u.6u(00 u); u(0) 500. (b) u 6u(400 u); u(0) 00. Annihilators What are the annihilators of the following functions? f t + 3t + 5. f sin(3t). f cos + 4 sin 4. f t cos(3t) 5. f t e t 6. f te 3t cos(6t) First Order Annihilator Problems 1

2 Solve the following nonhomogeneous problems using an integrating factor. This is the way we can find the solution prior to learning how to use annihilators. This is the technique that requires us to use integration by parts. (a) x x + 30; x(0). (b) x 1x + 6; x(0) 30.. Now solve the same nonhomogeneous problems using annihihilators. (a) x x + 30; x(0). (b) x 1x + 6; x(0) 30. Now we ll use the full methodology of the annihilator technique. For these problems tell me the (a) (b) (c) (d) (e) (f) (g) the homogeneous solution u h. the annihilator of the forcing function. the associated form of the particular solution u p. then solve the IVP. u 8u + 3t 3 + 4t + 0 u(0) 1 u u + e 6t u(0) 10 u 8u + 10e 8t u(0) 3 u 8u t + 10e 8t u(0) 30 u 3u + 0 u(0) 1 u 4u + 4t u(0) u u + 10e t u(0) 3

3 (h) u 3u + 10e 3t u(0) 30 Second Order Problems Homogeneous For these problems Find characteristic equation. Find operator equation. Solve IVP. y y + 10y 0 3 y (0) 1. y + 6y + 10y 0 4 y (0) y + 6y + 9y 0 3 y (0) 4 4. y + 6y + 8y 0 y (0) 6 5. y + 6y + 7y 0 7 y (0) 8 6. y + 4y + 4y 0 1 y (0) 1 7. y + 5y + 6y 0 y (0) 4 3

4 8. Second Order Problems Non Homogeneous For these problems Find characteristic equation. Find operator equation. Find homogeneous solution. y + 3y 10y 0 1 y (0) 40 Find particular solution using annihilator method. Solve IVP. If the roots are complex also write the solution in phase shifted form showing all work and provide a nice sketch y y + 10y 5t y (0) 1 y + 6y + 10y 8sin 4 y (0) y + 6y + 9y 6t 3 y (0) 4 y + 6y + 8y 9e 3t y (0) 6 y + 6y + 7y 4 cos 7 y (0) 8 y + 4y + 4y 10 sin(5t) + cos(3t) 1 y (0) 1 4

5 7. y + 5y + 6y 19te t y (0) 4 8. y + 3y 10y 6te t 1 y (0) 40 More Annihilators For the following problems tell me the form of y h tell me the annihilator of the forcing function tell me how many initial conditions would be needed to make an IVP. (a) (D )(D + 1)(D 6)y e 5t. (b) (D ) (D 1)y t. (c) (D ) (D 6)y e 6t. (d) (D ) (D + 4) y te 6t. (e) (D )(D + 1)y e 5t. (f) ((D + 4)(D 1)y e t. (g) ((D + 4)(D 1)y e 4t. (h) ((D + 4)(D 1)y. (i) (D 4D 6)y t 3. (j) (D + 8D + 8)y + t + e 6t. (k) (D 4D + 4)y t cosh t et +e t. (l) (D 5D 4)y 4e t + 5e 3t. (m) (D D + 1)y + 5t + 6t.. For the following forcing functions write down their annihilator. (a) f 4t + 6t + (b) f te t. (c) f te 8t. (d) f te t + e 8t + t 3 + t + 8. (e) f t e t. Eigenvalues and Eigenvectors: Find the eigenvalues and eigenvectors for A [

6 . Find the eigenvalues and eigenvectors for A [ Find the eigenvalues and eigenvectors for A [ Second Order Problems As Systems We will solve these as a system of ODE now. We didn t really do this in class but the ideas are straightforward. Something for you to remember if it comes up in the future in another class or in your work. For the following linear ODE Convert to a system of first order ODE Write down the characteristic equation. Find the general real and if necessary complexsolution Solve the IVP Do the phase plane analysis. 4. draw the z 1 0 line. draw the z 0 line. draw the eigenvector lines (distinct roots) draw the eigenvector line (equal roots) Divide the z 1 z into four regions corresponding to the algebraic signs of z 1 and z. Draw the trajectories of enough solutions for various initial conditions to create the phase plane portrait. y + 6y + 9y 0 4 y (0) 5 y + 7y + 6y 0 14 y (0) y + y 6y 0 7 y (0) y y 8y 0 1 y (0) 1 6

7 5. y y + 10y 0 3 y (0) 1 6. y + 6y + 10y 0 4 y (0) Linear Systems Write down the characteristic equation. Find the general real and if necessary complexsolution Solve the IVP Do the phase plane analysis draw the x 0 line. draw the y 0 line. draw the eigenvector lines (distinct roots) draw the eigenvector line (equal roots) Divide the x y into four regions corresponding to the algebraic signs of x and y. Draw the trajectories of enough solutions for various initial conditions to create the phase plane portrait. For the system below [ x [ x(0) [ [ 4 7 x 6 3 [ 3 1. For the system below [ x [ x(0) [ [ 4 10 [ x For the system below [ x [ x(0) [ [ 0 5 [ x 7

8 4. For the system below [ x [ x(0) [ [ 1 6 [ x 5. For the system below [ x [ x(0) [ [ 4 [ x 6. For the system below [ x [ x(0) [ [ 3 [ x 7. For the system below [ x [ x(0) [ [ 4 3 [ x Linear Systems with forcing functions For the system below [ x [ x(0) [ [ 4 7 x 6 3 [ [ For the system below [ x [ x(0) [ [ 4 10 [ x + [ 1 For the system below [ x [ x(0) [ [ 0 5 [ x + [ 1 8

9 4. For the system below [ x [ x(0) [ [ 1 6 [ x + [ 1 5. For the system below [ x [ x(0) [ [ 4 [ x + [ 1 6. For the system below [ x [ x(0) [ [ 3 [ x + [ 1 7. For the system below [ x [ x(0) [ [ 4 3 [ x + [ 1 Laplace Transformations Next we have the Laplace transform technique. First here are known Laplace Transforms. If F (s) L (f) then L (1) 1 s s > 0. L (t n ) n! s > 0. sn+1 L (e at f) F (s a). ( ) L (tf) d/ds F (s). Also L (e at ) L (cos(bt)) L (sin(bt)) 1 s a s > a. s s + b s > 0. b s + b s > 0. Then using the above we have 9

10 ( ) s L (t cos(bt)) d/ds s + b s > 0. s b (s + b ) s > 0. L (e at cos(bt)) L (te at cos(bt)) s a (s a) + b s > 0. (s a) b ((s a) + b ) s > a. ( ) b L (t sin(bt)) d/ds s + b bs (s + b ) s > 0. L (e at sin(bt)) L (te at sin(bt)) b (s a) + b s > 0. b(s a) ) s > a ((s a) + b If we define the step function H a by H a { 0 t a 1 t > a This is the same as the function H(a t) we have used. You should see different ways to write the same thing. Then L (H a ) e as /s s > 0. Next L (e bt H a ) e as /s with s replaced by s b. Thus L (e bt H a ) e a(s b) /(s b). Next noting { 0 t a H a f(t a) f(t a) t > a we have a nice way to handle H induced shifts: Finally we have shown that L (H a f(t a)) e as F (s) where F (s) L (f) L (δ(t a)) e as. 10

11 Solve the homogeneous problem u + 8 u + 1 u 0 u(0) 0 u (0) 6. Solve the homogeneous problem u + 16 u + 64 u 0 u(0) 10 u (0) Solve the homogeneous problem u + u + 17 u 0 u(0) 1 u (0) 3 4. Solve the nonhomogeneous problem u + 6 u + 5 u t 3 u(0) 13 u (0) 8 5. Solve the nonhomogeneous problem u + 6 u + 8 u f u(0) 1 u (0) 7 where f 6. Solve the nonhomogeneous problem { 3 0 < t < 3 4 t > 3 u u 15 u f u(0) 40 11

12 u (0) 3 where { 0 0 < t < 5 f 10δ(t 5) + 5 t > 5 7. Solve the nonhomogeneous problem u + u + 5 u 6 u(0) u (0) 3 8. Solve the nonhomogeneous problem u + 5 u e t u(0) 1 u (0) 4 9. Solve the nonhomogeneous problem u + 8 u + 0 u 6δ(t ) + 1 u(0) 6 u (0) Solve the nonhomogeneous problem u + 10 u + 9 u 0δ(t 3) + 50δ(t 6) u(0) 1 u (0) 1 1 Solve the nonhomogeneous problem u 3 u 4 u 10δ(t 3) + 7δ(t 5) u(0) 10 u (0) 7 1

13 Laplace Inversions Some inversions to do:. 4. 5/s + 6/s + 9e 5s /s s/(s + 9) 1e 3s /s e 8s /(s + 5) + 9e 11s s/(s + 49) e s + 1e 3s /[(s 1) e 7s s/[(s 6) + 9) 3e 6s /s [s 36/[s e 7s s/[s e 11s s/[(s 7) + 36) Predator-Prey Nullcline Analysis For the following predator systems find the (+ ) patterns for all 9 regions.. x x 4 x x x 9 x x x x 0 x 3 x 11 + x Predator-Prey Trajectory Analysis I For the following predator systems show the trajectories that start in the first quadrant stay there. Do this by using the linearization about the equilibrium point in the first quadrant. Draw a typical trajectory in Quadrant I. Also explain why these trajectories can t hit the x axis or y axis.. x 60 x 5 x x x x 1 x x x 7 x 8 x x 13

14 Predator-Prey Trajectories are bounded For the following predator systems show the trajectories are bounded in great detail.. Nonlinear ODEs For the following systems of nonlinear ODE find the equilibrium points x 60 x 5 x x x x 1 x x x 7 x 8 x x x f(x y) y g(x y) at each equilibrium point (x 0 y 0 ) analyze the local trajectories: find the Jacobian matrix at the critical point: [ f x (x f 0 y 0 ) y (x 0 y 0 ) g x (x g 0 y 0 ) y (x 0 y 0 ) Analyze the linearized system [ u v [ f 0 x f 0 y g 0 x g 0 y [ u v where u x x 0 and v y y 0. assemble the full picture x (x 4) (3y + 6) y ( 9x + y) (x 3). x (x 4) (y + 6) y ( 9 + y) (x 3) x (x 6) (3y + 6x) y ( 9x + y) (x 3) 14

15 4. x 3x 4xy y 5y + 1xy 5. x 3x 4xy x y 5y + 1xy y 15