3.3. SYSTEMS OF ODES 1. y 0 " 2y" y 0 + 2y = x1. x2 x3. x = y(t) = c 1 e t + c 2 e t + c 3 e 2t. _x = A x + f; x(0) = x 0.

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1 .. SYSTEMS OF ODES. Systems of ODEs MATH 94 FALL 98 PRELIM # 94FA8PQ.tex.. a) Convert the third order dierential equation into a rst oder system _x = A x, with y " y" y + y = x x x x b) The equation y " y" y + y = has general solution A y(t) = c e t + c e t + c e t Use this to nd a basis (x (t); x (t); x (t)) for the set of all solutions of _x = A x. The functions x j (t) should be vector valued. c) Show that the three solutions x j (t) you found in (b) are linearly independent. MATH 94 FALL 98 FINAL # 94FA8FQ.tex.. Determine the solution to the initial value problem where A = _x = A x + f; x() = x sin t ; f = cos t ; x = MATH 94 FALL 98 PRELIM # 94SP8PQ.tex.. Translate the following paragraph into a system of dierential equations in matrix form. Clearly dene all variables and constants. (Don't solve the equation). "The rate that the temperature of a cup of tea decreases is proportional to the dierence between its current temperature and the temperature of the room. The rate that the temperature of the room decreases is proportional to the dierence between the current temperature of the room and the temperature outside. The temperature outside is o C." (Note that the room is unheated and is unaected by the tea.)

2 MATH 94 SPRING 98 PRELIM # 94SP8PQabc.tex..4 Consider the system of ordinary dierential equations given by _x x _x = = = A x _x x a) Write this system in the form ay" + by cy = (Hint: y = x ; y = _x = x ) b) Solve the second order equation above by any method you want. c) Use your solution to part b above to nd a basis for the vector space of solutions x x = of _x = A x: x MATH 94 SPRING 98 PRELIM # 94SP8PQ.tex..5 Consider the system of equations given by d x x = + f(t): x x a) Find the general solution to the above system when f(t) =. b) Find the fundamental matrix solution e A(t) to the above system (still with f (t) = ) c) Using any method, solve the above system when f(t) = e t and x() = MATH 94 SPRING 98 MAKE UP PRELIM # 94SP8MUPQ.tex..6 Consider the equation _x A x = f(t) (),where 4 A = 4 a) Find the general solution to (*) with f(t) = b) Find e A(t) e t c) for f(t) = and the initial condition x() = MATH 94 SPRING 98 FINAL # 94SP8FQ.tex..7 Solve for x(t) by any method: _x = A x + f(t); A = sin(t) ; f(t) = ; x() =

3 .. SYSTEMS OF ODES MATH 94 SPRING 984 FINAL # 4 94SP84FQ4.tex..8 Find the general solution of y" + ay + by = as follows: Let z = y and z = y. Write the second order dierential equation as a system. Solve the system and then convert the answer back in terms of y. Compare the result with the general solution of y" + ay + by = found using any other technique. Hint: If is an eigenvalue of a x matrix a a a a a a, then the eigenvector is MATH 94 SPRING 984 FINAL # 6 94SP84FQ6.tex..9 Find a solution of _x(t) = Ax + f by "expanding" x(t) = P a(t) and solving for a(t). P is the diagonalizing matrix for A, ie., A = P DP. Here x(t) = and 4 x (t) x (t) x (t) 5 ; A = 4 a = 4 a (t) a (t) a (t) 5 ; x() = 4 5 ; f(t) = 4 t MATH 94 FALL 984 FINAL # 94FA84FQ.tex.. a) Find the eigenvalues and eigenvectors of the matrix A b) Find A and A T. c) Find the general solution of d dx y y y A A A y y y A : MATH 94 SPRING 985 FINAL # 94SP85FQ.tex.. For the case ~ b 6= ~, the vector ~x = ~ a) Is always a solution. b) May or may not be a solution depending on A. c) Is always the only solution. d) Is never a solution. 5

4 4 MATH 94 FALL 985 FINAL # 6 94FA85FQ6.tex.. Find the general solution of the system Y = AY where Y = dy 5 ; and A = : MATH 94 FALL 986 FINAL # 5 94FA86FQ5.tex.. Solve the system of dierential equations Y = AY,where Y is the vector A is the matrix : MATH 94 SPRING 987 PRELIM # 7 94SP87PQ7.tex..4 The dynamics of the systems below can be put in the form: _x = [A]x y ; Y = y y ; and y Using x as dened in one of the problems below, nd the appropriate matrix [A]. Q a) Electric circuit: x = (Q is capacitor charge, i is current) i x b) Mechanical system: x = v (x = position, v = velocity)

5 .. SYSTEMS OF ODES 5 MATH 94 SPRING 987 PRELIM # 5 94SP87PQ5.tex..5 Find the value of x at t = if _x = x; with x() = MATH 94 SPRING 987 PRELIM # 94SP87PQ.tex..6 Find the general solution of _x = x: MATH 94 SPRING 987 FINAL # 94SP87FQ.tex..7 Pick any value (a specic real or complex number) so that you can nd a real non-zero solution to the equations below and then nd any non-zero solution. If no such exists, say so. a) x + x = with x() = b) x + x = with x() = and x () = : c) x + x = with x() = and x(7) = d) x = x MATH 94 FALL 987 PRELIM # 6 94FA87PQ6.tex..8 Find the general solution of x' = Ax; where A = MATH 94 FALL 987 MAKE UP PRELIM # 6 94FA87MUPQ6.tex..9 Find the general solution of x' = Ax, where A = 4 MATH 94 FALL 987 PRELIM # 94FA87PQ.tex.. Find the general solution of the system x' = Ax: where A = : 7 5 & x = 6 4 x (t) x (t) x (t) x 4 (t) :

6 6 MATH 94 FALL 987 MAKE UP PRELIM # 94FA87MUPQ.tex.. a) Find the general solution of the system _x = Ax if A = C A and x = b) How many independent eigenvectors can we nd for the matrix C A? MATH 94 FALL 987 FINAL # 6 94FA87FQ6.tex x x x x 4.. Consider x' = Ax, where A is a x matrix. If P = 4 C A 4 5 and P AP = 5 : nd the general solution of the system of dierential equations. MATH 94 FALL 987 MAKE UP FINAL # 7 94FA87MUFQ7.tex.. Solve x' = Ax if A A and x x x MATH 94 SPRING 988 PRELIM # 6 94SP88PQ6.tex..4 Translate the following situation to a system of rst order Ordinary Dierential equations in standard matrix form. Choose arbitrary values for any proportionality constants. x; y and z are functions of time t. x increases at a rate proportional to y. y increases at a rate proportional to z. z increases at a rate proportional to x + y + z. x : MATH 94 SPRING 988 PRELIM # 7 94SP88PQ7.tex..5 Find so that x(t) = e t 4 A 5 dx satises the equation = 4 5 x

7 .. SYSTEMS OF ODES 7 MATH 94 SPRING 989 PRELIM # 4 94SP89PQ4.tex..6 Consider the system of equations dx = x + y dy = x y dz = 6y + 4z a) Write the system in the form Dx = Ax b) Find all the eigenvalues of A and their corresponding eigenvectors. c) Let ; ; and be the eigenvalues, and v ; v ; and v the corresponding eigenvectors found in Part b. Using the Wronskian, show that x(t) = c e t v + c e t v + c e lambda t v is the general solution to the given homogeneous system d) Consider the nonhomogeneous system Dx = Ax + E(t) with A as the matrix from Part a, and E(t) = 4 t t 6t 5 ; A particular solution of this system is given by p(t) = 4 t 5 : Write down the general solution to the nonhomogeneous system. MATH 94 SPRING 989 PRELIM # 5 94SP89PQ5.tex..7 Consider a homogeneous system of dierential equations, Dx = Ax whose general solution is given by x(t) = c 4 et e t e t 5 + c 4 e t e t 5 + c 4 e t e t Find the specic solution which satises the initial condition given by, x () = ; x () = ; x () = : That is, nd the c's such that x(t) = c h (t) + c h (t) + c h (t) at t = 5

8 8 MATH 94 SUMMER 989 PRELIM # 94SU89PQ.tex..8 a) Write d x d x dx + x = as a linear system of ordinary dierential equations. Dx = Ax b) Find the general solution x(t) to the above system. c) Find a basis for the solution space of the system and show that the vectors constituting this basis are linearly independent. MATH 94 FALL 989 PRELIM # 94FA89PQ.tex..9 Consider the x system of dierential equations, (I) Dx(t) + Ax(t); A = : a) Verify that h (t) = e t and h (t) = e t are two linearly independent solutions of (I). [For part b and c below, you may use the result of part a even if you did not show it] b) Find a particular solution for c) Taking on faith that Dx(t) = Ax(t) + t + t (II) Dx(t) = Ax(t) + is a particular solution of ; t nd the solution of (II) that satises the initial condition x() =

9 .. SYSTEMS OF ODES 9 MATH 94 FALL 989 PRELIM # 94FA89PQ.tex.. Consider the system of ordinary dierential equations where A = Dx = Ax; 4 5 : a) Find the general solution in terms of real-valued functions b) Find the solution satisfying the initial conditions x() = 4 MATH 94 FALL 989 FINAL # 94FA89FQ.tex.. Consider the homogeneous system of dierential equations where 5 (*) Dx = Ax; A = 4 5 a) Show that = is a double eigenvalue of A and that = is a simple eigenvalue. In part b below, you may use the result of part a even if you did not show it. b) Find the general solution of (*). MATH 94 FALL 989 FINAL # 94FA89FQ.tex.. Consider the inhomogeneous system of dierential equations of x (t) and x (t). dx = x x + e t : dx = x + x a) Find the real-valued general solution of the corresponding homogeneous problem. b) Find a particular solution for the given inhomogeneous problem. c) Find the solution of the inhomogeneous problem which satises the initial condition x () = ; x () =

10 MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex.. Find the solution of the initial-value problem subject to dx dx = x + x = x + x x () = ; x () = 5 MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..4 Find the general solution of x = x x = 5x x + e t x + 5 sin t MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..5 Solve the initial-value problem x = 4x x x = 4x x () = ; x () = : MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..6 Find the general solution x = Ax, where A = Hint: the eigenvalues of A are -4,, 5 and x x x x MATH 94 SPRING 99 FINAL # 94SP9FQ.tex..7 Find the general solution of x = x + t; x = x + x A

11 .. SYSTEMS OF ODES MATH 94 SPRING 99 FINAL # 8 94SP9FQ8.tex..8 Find the general solution of x" = Ax; where Hint: A 4 A is an eigenvector of A MATH 94 SUMMER 99 PRELIM # 94SU9PQ.tex..9 Consider the following system of dierential equations" 5 and dx = 4x + 6y + g(t) dy = x y: a) Find the complementary solution, that is, the solution of the homogeneous system with g(t) =. b) Find the particular solution for g(t) = + e t MATH 94 SUMMER 99 PRELIM # 94SU9PQ.tex..4 Find the general solution of the following second order system: d x = 4x ; and d x = x + 4x + x ; d x = x

12 MATH 94 SUMMER 99 PRELIM # 4 94SU9PQ4.tex..4 In the system shown, x, is the displacement of mass m; x is the displacement of mass m. When x x = the spring is relaxed. The spring constant is k. Under the usual assumptions (no friction; mass of spring relatively small, etc...), a) derive the equations of motions for the system, and put your solution in the form of a x second order homogeneous system; b) show that the eigenvalues for this system are = and = k( m+m m m ); c) explain the physical meaning of the zero eigenvalue. What could be added to the system to get rid of the zero eigenvalue? MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..4 Determine the analytic solution of dy dx = y sinx; y() = MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..4 Determine the solution to the initial value problem dx = x + x + e t ; x () = dx = x + 4x + 6; x () = Hint: the eigenvectors are (,) and (,) MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..44 Determine the most general solution of the system dx = 4x 5x dx = x + x Hint: the eigenvalues are i. You may express your solution as the sum of two complex conjugate functions.

13 .. SYSTEMS OF ODES MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..45 For the system below, write down all of the algebraic equations you would have to solve to determine a particular solution: dx = 4x 5x + 4 sin t dx = x + x + + t: MATH 94 FALL 99 PRELIM # 4 94FA9PQ4.tex..46 Consider a physical system consisting of two identical masses and three identical springs The equations of motions are mx" = kx + k(x x ) mx" = k(x x ) kx : a) Express the most general solution of this system in terms of sines an cosines. Hint: =-(k/m) and -(k/m) b) Identify the displacements associated with each normal mode. MATH 94 FALL 99 FINAL # 4 94FA9FQ4.tex..47 Determine the analytic solution of the initial value problem x" =! x +! x +! x x' () = x () = x" =! x! x x' () = x () = x" =! x +! x x' () = x () = Hint: The eigenvectors are (,,-), (,,), and (-,,).

14 4 MATH 94 FALL 99 MAKE UP FINAL # 4 94FA9MUFQ4.tex..48 Determine the analytic solution of the initial value problem x" =! x +! x +! x x' () = x () = x" =! x! x x' () = x () = x" =! x +! x x' () = x () = Hint: The eigenvectors are (,,-), (,,), and (-,,). MATH 94 SPRING 99 PRELIM # 4 94SP9PQ4.tex..49 a) Consider the initial value problem d y dx + xy x dy x = with y() = and dy dx dx () = b) Write the equation as a rst order system. c) Use Euler's method with a step size of 4 on the system to obtain the approximate solution for y at x = MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..5 Consider the inhomogeneous system dx = x + x + e t dx = 5x x + a) Determine the general solution of the homogeneous system. b) Determine a particular solution c) Determine the solution of the inhomogeneous system that satises the initial conditions x () = x () = 6 5 MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..5 Find a particular solution of the system dx = dy x + y + cos t = x + y:

15 .. SYSTEMS OF ODES 5 MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..5 Consider the second order system x" = x + y + z y" = x y z" = x z a) Write down the corresponding characteristic equation and show that its roots are, -, and -. b) Determine the general solution of the homogeneous system and express it in terms of real-valued functions. c) Determine the circular frequency of the normal mode that the initial conditions x() = ; y() = z() = ; x () = y () = : will excite. MATH 94 SPRING 99 FINAL # 4 94SP9FQ4.tex..5 Express the general solution of the system d x = x + y and d y = x y In terms of real-valued functions. Then determine the solution that satises the initial conditions x() = ; y() = ; dx dy () = ; () = : Hint: The eigenvectors of the matrix of coecients are (,) and (, ) MATH 94 SPRING 99 FINAL # 5 94SP9FQ5.tex..54 Find a particular solution of d x x + e t : Hint: The characteristic equation factors to ( )( + + ) = : a) Determine the unique solution of _x = Ax, where A = ; with x() = :

16 6 MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..55 Consider the system of rst order equations dx = dy = x x + y t + z + et dz = y z et Given that the eigenvectors of the coecient matrix are (,,), (,, -) and (,-,)., a) Determine the general solution of the homogeneous system. b) Determine a particular solution. c) Determine the solution of the inhomogeneous system that satises the inital conditions x() = ; y() = ; z() = : MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..56 Determine the general solution of the second order system x" = x + 7y y" = x y and express it in terms of real-valued functions. MATH 94 FALL 99 FINAL # 4 94FA9FQ4.tex..57 Consider the inhomogeneous system dx = 4x x + e t dx = 5x x + sin t a) Determine the general solution of the homogeneous system. b) Determine a particular solution. c) Determine the solution of the inhomogeneous system that satises the initial conditions x () = x () = MATH 94 SPRING 99 PRELIM # 4 94SP9PQ4.tex..58 Describe the following situation by a set of dierential equations: "A barrel of water has temperature T b (t). It sits in a room with temperature T r (t). Outside the room the temperature is a constant o F. The barrel cools at a rate proportional to the dierence between its temperature and the room temperature. The room cools due to heat ow to the outside at a rate proportional to the dierence between the inside and outside temperature. The room is simultaneously heated by the barrel at a rate proportional to the dierence between the barrel temperature and the room temperature." Clearly dene the constants and state which are positive in your description.

17 .. SYSTEMS OF ODES 7 MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..59 Consider the system of dierential equations dx = x + y dy = 6x + y a) Find the general solution. b) Find the solution which satises the initial conditions x() = and y() =. MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..6 Consider the initial value problem d y ty dy dy + y = t; y() = and () = a) Rewrite this problem as an initial value problem for a rst order system of two dierential equations. b) Write down Euler's method for the system in part a) using step size h. c) Use part b) with h =. to calculate the approximate solution at t =. and t =. MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..6 Find a particular solution of the system dx = x et dy = z + sin t dz = y + 5 Note: You are not asked to nd the general solution of the system. MATH 94 SPRING 99 PRELIM # 4 94SP9PQ4.tex..6 Given the vector eld F(x; y) = x i + xyj and the closed curve C shown below, compute: a) the (counter-clockwise) circulation of F around C; b) the ux of F across C. MATH 94 SPRING 99 PRACTICE PRELIM # 94SP9PPQ.tex..6 Find a particular solution of the system dx = x + y + cos t + dy = t x + y + e

18 8 MATH 94 SPRING 99 FINAL # 4 94SP9FQ4.tex..64 Find the general solution of dx = x dy y + sin t + cos t = x y + : MATH 94 SPRING 99 FINAL # 5 94SP9FQ5.tex..65 a) Find the general solution of d x = x + y d y = 4x y: Write your solution in terms of real valued functions. b) Find the solution satisfying initial conditions x() = ; y() = ; x () = y () = : MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..66 a) Find the general solution of x = + e t 4 ; where x = x : x b) Find the solution satisfying the initial condition x() =. MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..67 Consider a second-order system, x" = Ax; where A is a x symmetric matrix, and x is -vector. Although A is not specied, it is known that A A : Find the solution satisfying the initial conditions x() = MATH 94 FALL 99 FINAL # 94FA9FQ.tex..68 Find the general solution of x = x + x + : x = 4x + A ; x'() = :

19 .. SYSTEMS OF ODES 9 MATH 94 FALL 99 FINAL # 6 94FA9FQ6.tex..69 Find the eigenvalues and eigenfunctions (nontrivial solutions) of the two-point boundary-value problem y" + y = ; < x < ; ( assume ) y () = y() = : MATH 94 FALL 99 FINAL # 94FA9FQ.tex..7 Solve the initial-value problem x" = Ax (where x is a -vector), x() 4 A ; x () = ; where A = 4 4 Hint: If you think they are required, all eigenvalues/eigenvectors can be found as follows: one eigenpair follows by inspection; the other two eigenpairs were essentially computed in problem (see. Q68) MATH 94 SPRING 99 PRELIM # 94SP9PQ.tex..7 a) Find the general solution of x = Ax if A = 4 b) Find the particular solution with x() A : MATH 94 SPRING 99 PRELIM # 4 94SP9PQ4.tex..7 A damped spring-mass system satises u" + u + u =. Dene x (t) = u(t); x (t) = u x (t); x(t) = : x a) Show that x satises a system x = Ax and nd A. b) Find the general solution of the system in (a). MATH 94 SPRING 99 MAKE UP PRELIM # 94SP9MUPQ.tex..7 Solve x x() 5 5 A x A 5

20 MATH 94 SPRING 99 MAKE UP PRELIM # 4 94SP9MUPQ4.tex..74 a) If you were to try to solve x = x + e 5 4 iwt by substituting x(t) = ve iwt ; what algebraic equation would v have to satisfy? (You are not asked to solve it.) b) Give an example of a 4x4 matrix all of whose eigenvalues are real. c) If x = x + ie it and r(t) is the real part of x(t), what rst order ODE does r(t) solve? (You are not asked to solve it.) MATH 94 SPRING 99 FINAL # 94SP9FQ.tex..75 Find the general solution of x 4 = x: 4 MATH 94 FALL 99 PRELIM # 5 94FA9PQ5.tex..76 Find a second order dierential equation which x (t) solves whenever x = x: Here x is the second component of the vector x. MATH 94 FALL 99 MAKE UP PRELIM # 5 94FA9MUPQ5.tex..77 Reduce u" + :5u + u = sin t to a system of rst order equations and express in matrix form. MATH 94 FALL 99 PRELIM # 94FA9PQ.tex..78 a) Find the eigenvalues and eigenvectors of the system x! = x;! b) Find the general real-valued solution for the system in part a. c) Sketch the phase portrait. Are these solutions stable? d) Write the second-order dierential equation that corresponds to the system in part a.

21 .. SYSTEMS OF ODES MATH 94 FALL 99 PRELIM # 4 94FA9PQ4.tex..79 a) Find the solution of X 4 = X 4 7 for X() = : b) The phase portraits near equilibrium for three systems of two st order equations are sketched below. Match the portraits with the eigenvalue pairs listed. (i) = + i; = i (ii) = + i; = i (iii) = ; no (iv) = ; = (v) = ; = (vi) = ; =

22 MATH 94 FALL 99 FINAL # 94FA9FQ.tex..8 a) Solve y"(t) + y (t) + y(t) = ; y() = ; y () = b) Put the equation in (a) into the form x = Ax; where x = y; x = y. Put the initial conditions in vector form too. Sketch several trajectories in the phase plane. c) For the forced equation y" + y + y = coswt; nd a value for w and a particular solution such that the friction term y matches the force coswt, while the acceleration y" and spring force term y cancel each other. MATH 94 SPRING 994 FINAL # 6 94SP94FQ6.tex x (t)..8 Suppose that x(t) = and x() = : Then nd x(t) for the following x (t) systems of ODE's: a) x = x b) x t = t c) x = x Hint: It may help to write the ODE's for x and x separately in some cases. MATH 94 FALL 994 PRELIM # 94FA94PQ.tex..8 Solve the following a) y" + y + y = ; y() = ; y () = : b) y = y cosx, nd general solution. MATH 94 FALL 994 PRELIM # 94FA9PQ.tex..8 Given y(t) = e t sin(t), a) Find an ordinary dierential equation (y" + by + cy = ), solved by y. b) Find an equivalent system of the form x = Ax. c) Sketch the phase portrait of the system of equations. Show the curve corresponding to y as a curve on the phase plane. d) Solve the system x = Ax if x() =

23 .. SYSTEMS OF ODES MATH 94 SPRING 995 FINAL # 94SP95FQ.tex..84 Let x 6 x = y y a) Write a fundamental set of solutions to the system. b) Give the solution x(t) y(t) to the system such that x() = ; and y() = : c) Sketch your solutions x(t) and y(t) to part (b) in the x t; and y t planes. d) Sketch the phase plane for this system showing characteristic trajectories, including that of your solution to (b) (indicate it), with arrowheads indicating direction of movement as t!. e) Transform this system of equations to a second order equation in x plus a rst order equation in x and y that may be solved successively to obtain the same solutions as the original system.

24 4 MATH 94 SPRING 995 FINAL # 94SP95FQ.tex..85 Consider the system x a b x = y c d y Suppose the phase plane picture of trajectories looks like this: a) If x(t) y(t) is a solution with x() > and y() > ; sketch the graphs of x(t) and y(t) for < tm: b) Same as (a) of x() < ; y() <. c) What information can you infer about the matrix a b c d from the phase plane picture? d) Give an example of a matrix such that the system will have this phase plane picture. MATH 94 SUMMER 995 QUIZ # 94SU95QuizQ.tex..86 a) Convert y" + y + 5y = to a system of rst order dierential equations. b) Write the above in a matrix form, i.e. x = Ax

25 .. SYSTEMS OF ODES 5 MATH 94 SUMMER 995 QUIZ # 94SU95QuizQ.tex..87 a) Solve the system of linear, rst order dierential equations, x = x; with x() = 4 b) Sketch the phase plane portrait corresponding to the above system of dierential equations. c) Show the solution corresponding to the given initial condition on the phase plane portrait. d) Sketch the solutions x and x vs t, i.e. x (t) and x (t) for the above I.C. MATH 94 FALL 995 PRELIM # 94FA95PQ.tex..88 Write a system of st order equations which is equivalent to the equation y" + y + y + y =. Don't try to solve. MATH 94 FALL 995 PRELIM # 94FA95PQ.tex..89 For each of the systems nd the general solution and sketch the phase plane. Also show the solution passing through x() = ; y() = both on your phase plane sketch and on plots of xvst and yvst with as much detail as you can give. Note that you are given most of the eigenvalue and eigenvector information you need. x, repeated eigenvalue, one linearly indepen- x a) y = y + i given i i ( i) i x 4 x b) y = 4 7 y 4 given 4 7 i dent eigenvector x x c) y = 4 y given = 4 + i = ( +i) i = i and i i and 4 MATH 94 FALL 995 FINAL # 94FA95FQ.tex..9 This concerns the system of ODE's x = Ax; where A = a) Find the general solution. b) Solve for x(t) if x() = t! = 5 ; and describe the behavior of this solution as =

26 6 MATH 94 SPRING 996 PRELIM # 94SP96PQ.tex..9 Solve the system x = x + y y = x y : with the initial conditions x() = 4; y() = 5: You may use the fact that the associated matrix has eigenvalues + i and i. Sketch the solution curve in the (x; y) plane. Also nd the critical point (i.e. equilibrium or constant solution) of this system. MATH 94 SPRING 996 MAKE UP FINAL # 4 94SP96MUFQ4.tex..9 Consider the system x = x () a) Write the corresponding second-order ordinary dierential equation. b) Find the eigenvalues of () and the character of the critical point(s). c) Find the eigenvalues for x = x () What is the character of the critical point(s) of (), and how does the stability depend on the sign of? MATH 9 FALL 996 PRELIM # 9FA96PQ.tex..9 Find y() given that x() = ; y() = ; and: _x = x + 4y _y = 4x + y MATLAB option. You can get full credit for this problem by writing a set of MATLAB commands and/or functions and/or script les which would give y(). You can get full credit by solving by hand also.

27 .. SYSTEMS OF ODES 7 MATH 94 FALL 996 PRELIM # 94FA96PQ.tex..94 a) Write the system dx = x where x = x x : as a second-order ordinary dierential equation. b) Find the eigenvalues for the problem given in part a. Locate the critical point(s) of this system and describe its (their) character. Sketch a few trajectories in the phase plane. c) Find the eigenvalues of dx = x; << What is the character of the critical point(s) of this equation and (how) does the stability depend on 's sign? Sketch a few trajectories in the phase plane for <, and plot x (t) and x (t).

28 8 MATH 9 FALL 996 PRELIM # 4 9FA96PQ4.tex..95 No partial credit. You need correct reasoning and the correct answer to get credit for each part. No credit for part(b) unless part(a) is correct. Consider the pair of coupled ordinary dierential equations: dx = x(5 y) dy = y(5 x) a) What are the critical points for these equations? b) One and only one of the three pictures below (made by PPLANE) is the vector eld for the equations above. Pick and use the correct picture to determine for each critical point whether it is stable or unstable.

29 .. SYSTEMS OF ODES 9 MATH 94 FALL 996 MAKE UP PRELIM # 94FA96MUPQ.tex..96 a) Show that x = t r ; where a constant vector and r a constant, will give non-trivial solutions to tx = Ax if and r satisfy (A ri) = : b) Find a general solution for the system of equations, assuming t > tx 4 = x: 8 6 MATH 9 FALL996 FINAL # 9FA96FQb.tex..97 dx = x with x() = 7 % This is a diary of a MATLAB session" % It does all the calculations needed for this problem >> [V,D] = eig([ ; ] ); >> V = sqrt() * V ; % this tidies up V >> [V,D] ans = >> V/[ 7]' ans = - 5 MATH 94 SPRING 997 PRELIM # 94SP97PQ.tex..98 You are told that for a certain matrix B it is true that B = and B : Solve the initial value problem x = Bx x() = : x You are not asked to nd B. (Notation: x = x = x = and = x d ) =

30 MATH 94 FALL 997 PRACTICE PRELIM # 94FA97PPQ.tex..99 Consider the matrix A = 4 5 () It is given that the characteristic equation of A is: ( ) = a) Find all the eigenvalues of A. b) Find all possible linearly independent eigenvectors of A. c) Is A diagonalizable? (i.e. Is A similar to a diagonal matrix D? ) If so, write down a diagonal D that is similar to A. Now consider the system of ordinary dierential equations: with x = Ax () x() = 4 5 (4) where x(t)r and x = dx d) Find a general solution to () e) Find the unique solution to x(t) of (,4). f) Verify that x(t) satises (,4). MATH 94 SPRING 998 PRELIM # 6 94SP98PQ6.tex.. A certain real x matrix A has the following properties: A 4 5 = 4 5 A 4 i + i i 5 = 4 i i i Without nding the matrix A, nd the real form of the general solution to x = Ax. MATH 94 SPRING 998 FINAL # 5 94SP98FQ5.tex.. Construct the general solution of x = Ax involving complex eigenfunctions and then obtain the general real solution when A = : 5

31 .. SYSTEMS OF ODES MATH 9 FALL 998 PRELIM # 4 9FA98PQ4.tex.. Determine the general solution of the sytem x = Ax; when A = 4 5 : Hint: The eigenvalues are -, -, and 5. MATH 9 FALL 998 PRELIM # 5 9FA98PQ5.tex.. Determine the solution of the system x = Ax; where A = ; 8 that satises the initial conditions x() = : MATH 9 FALL 998 FINAL # 7 9FA98FQ7.tex..4 Determine the solution of the initial-value problem where A x = Ax; x() = x ; 4 5 and x 4 5 : MATH 9 FALL 998 FINAL # 8 9FA98FQ8.tex..5 Give the general solution, in terms of real valued functions, of the system of dierential equations x = x: