Entrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
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1 Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the solution y(x) satisfies Prove your conclusion.. Solve the initial value problem lim y(x) =. x { y 3y =, y() =, y () = Consider the following second order differential equation: Let f(t) be a function satisfying y + ( + sin t)y ( + t ) y =. f() =, f() =, f() =. Is it possible that f(t) is a solution? Prove your conclusion. 4. Given a solution y = t (t > ) for the second order differential equation Find the general solution for t y t(t + )y + (t + )y =. t y t(t + )y + (t + )y = t 3, (t > ) 5. Solve ẋ = Ax = 3 x, x() = 3 Find e At for this problem. Write down the solution formula for ẋ = Ax + f(t), x() = 3
2 6. Consider ẋ = y + x y, ẏ = x xy + x 3, where (x, y) R. Show that the system is Hamiltonian. Find the Hamiltonian function. Find the critical points and name them. Also, discuss their stability. Sketch the phase diagram. Be sure to put Arrows in orbits (or paths) to indicate the direction of time. 7. For the following system in R : ẋ = x y x(x + y ), ẏ = x + y y(x + y ). (a) Discuss the stability of critical points. (b) Show if there is a periodic solution. 8. Consider ẋ = f(x), () where x R n, f is continuously differentiable, f() =, and x = is an isolated critical point. Suppose that the linearized system is given by where is the Jacobian of f evaluated at. ẋ = Ax, () A = f x () Suppose the zero solution of () is stable. Either prove that the zero solution of () is stable or show a counter example.
3 Differential Equations Exam, 6S. Solve five problems. [] Let p(t) and q(t) be two continuous functions. Prove that the second order linear equation y + p(t)y + q(t)y = has two, and only two linearly independent solutions. [] Consider the second order differential equation : y (sin t)y + ( + cos t)y =. It is easy to see that y = sin t is a solution with period π. Are all the solutions must be periodic with period π? Prove your conclusion. [3] Prove Gronwall inequality: Let k(t), u(t) be continuous in [a, b] with k(t), and C is a constant. If t u(t) C + k(s)u(s)ds t [a, b], a then t u(t) Ce k(s)ds a t [a, b]. [4] (a) Solve ẋ = 4 4 x, x() = (b) Find a Jordan (canonical) form J of the above matrix and the transformation P such that P AP = J. (c) Discuss the stability of the system for t. [5] Consider ẋ = + e +t t ln(t+) +t ẋ = x xy + x 3, ẏ = y xy. 3 x + e 3t te t (a) Find the critical points and name them. Also, discuss their stability. (b) Along what curves (or lines), is dx zero or infinite? (c) Are the following sets invariant? Circle the invariant sets. () The line y =. () The line x =. (3) The line x =. (4) The curve y + x =. (d) Sketch the phase diagram. Be sure to put Arrows in orbits (or paths) to indicate the direction of time. [6] Discuss the stability of critical points. (a) (b) ẋ = y 3 + x y, ẏ = xy + x 3. ẋ = x xy, ẏ = x 4 y y 3.
4 Differential Equations Exam, 4F. Show and explain all work. Do 6 out of the 7 problems. [] Consider the planar system dx = x + y x 3, = x + y y 3. (i) Determine if the origin is stable, asymptotically stable, or unstable. (ii) Determine if there is a periodic solution. Either case, show your proof. [] Consider the planar system dx = y + x[µ ( x y ) ]( x y ), = x + y[µ ( x y ) ]( x y ), where µ is a parameter independent of t, x, and y. (i) Rewrite the planar system in polar coordinates. (ii) Show that there is a pitchfork bifurcation of a periodic orbit as the parameter µ is changed. (iii) Sketch the phase diagrams according to the values of µ. [3] Consider the planar system where a and b are positive constants. dx = y axy, = x by, (i) Find all critical points and classify them according to stable, asymptotically stable, and unstable. (ii) Determine the ω-limit set and α-limit set for every point in the plane. If there is no such set, state so. [4] (i) Determine a fundamental solution to the system of ODE s = y, < t <. () (ii) Are there non trivial (non zero) column vector solutions of () that converge to t? If yes determine all of them. (iii) Solve the initial value problem = y +, y() =. as
5 [5] Let Y = Y (t) be an n by n matrix solution Y (t) of dy = A(t)Y, where A(t) = [a jk (t)], j, k =,...n, and A(t) is an n by n continuous matrix function on (a, b). Denote by q(t) the determinant of Y (t). (i) Show that q(t) is a solution of the first order homogeneous ODE dq(t) n = [ a jj (t)]q(t). () j= (ii) Determine a particular solution q(t) of () that satisfies the initial condition q(t ) = q, with t (a, b). (iii)show that if Lim t b [ t t [ n j= a jj (s)]ds] =, then Lim t b [q(t)] =. (iv) What is the significance of the solution q(t)? [6] Let A be an n by n constant matrix with eigenvalues λ, λ,..., λ n such that Re{λ j (t)} <, j =,...n. Let y(t) and h(y) be column vectors given by y T (t) = (y (t), y (t),..., y n (t)) and h T (y) = (y, y,, y n). Show that the zero vector solution of is asymptotically stable as t. = Ay + h(y) [7] (i) Determine an interval of existence for the solution of the initial value problem = y + y +, y () =, = y + y +, y () =. (ii) Explain what theorem guaranteeing existence and uniqueness you apply. (iii) Show that the initial value problem t = y, y() = possesses infinitely many solutions. Then explain why a theorem guaranteeing the existence of unique solutions does not apply to this initial value problem.
6 Differential Equations Exam, 3F. Solve all five problems. [] (a) Solve ẋ = x, x() = The characteristic polynomial of the above coefficient matrix is λ (λ ). (b) Find a Jordan form J of the above matrix and the transformation P such that P AP = J. (c) Find the stable, center, and unstable subspaces if there are. [] Consider ẋ = x + x, ẋ = x + x. (a) Find an approximation for the stable, center, unstable manifolds at the critical point x =. If a particular manifold does not exist, state so. (b) Draw the phase diagram near the origin including the above manifolds. (c) Is the critical point x = stable or unstable? [3] Discuss the stability of critical points. (a) (b) ẋ = x3 + xy, ẏ = y 3. ẋ = 4x y, ẏ = x + xy 4. 3 [4] Consider ẋ = y, ẏ = x + y( x y ). (a) Show the existence of a periodic orbit. (b) Assuming that the periodic solution (a) is unique, show the stability of the periodic orbit. [5] (a) Derive the variation of constant formula for a system ẋ = A(t)x + f(t), where A(t) is an n n matrix and x, f R n. Assume that a matrix solution X(t) to ẋ = A(t)x is given. Hint: X (t) (the inverse of X(t)) satisfies (Ẋ ) = X A(t), where (Ẋ ) = d (X ). (b) Find the solution of ẋ = x + e t, x() = The characteristic polynomial of the above coefficient matrix is (λ ) 3. 3
7 Ph.D. Entrance Exam Differential Equations May 7, 996 Part I: Ordinary Differential Equations Instruction: Complete 4 of the 5 problems.. Find and classify all equilibrium points of the system Sketch the phase diagram of the system. ẋ = 6y + xy 8, ẏ = y x.. For the autonomous system ẋ = X(x, y), ẏ = Y (x, y), suppose X and Y satisfy the following conditions: X(, ) = Y (, ) =, X x > X y and Y y > Y x in a neighbourhood of (, ) Use V (x, y) = max{ X(x, y), Y (x, y) } as a Liapunov function to determine the stability of the zero solution. 3. For the second order equation ẍ + ɛ( x + ẋ )ẋ + x = where ɛ is a positive parameter, (a) Show that there exists at least one periodic solution. (b) Suppose x(t) is a periodic solution with period T. Show that x(t) has exactly two zeros in [, T ). (c) When the parameter ɛ > is small, find the approximate amplitude a of the periodic solution by using the energy balance method, and determine whether it is stable. 4. For the equation ẍ + f(x) =, where f(x) = sgn(x) when x > and f(x) = x when x, (a) show that all solutions are periodic solutions. (b) Find the amplitude and period of the solution that satisfies the initial conditions x() = and ẋ() = b >. (c) Obtain the explicit expression for the solution x(t) in (b). 5. Show that the boundary value problem has a unique nontrivial solution. ẍ + x = ( < t < ), x() = x() =
8 Some useful integrals: π/ sin k t = π/ cos k t = k = π/4 k = /3 k = 3 3π/6 k = 4
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