ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is this limit the same as the solution of the equation Lẏ + Ry = 0? 2. (a) Solve ẋ = x x 2 with initial condition x(0) = x 0. (b) Consider x 0 [0, ). Does the solution exist for all t [0, )? (c) Consider x 0 (,0). Does the solution exist for all t [0, )? (d) Does the solution exist for t ( infty,0] for each of the above cases? (e) Is there a finite T > 0, independent of x 0 such that the solution exists for all t [0, T]? 3. Problems 7,8 of page 42 of Coddington 4. Problem 7 of page 46 of Coddington 5. Find at least one (non-trivial) solution of the following equations. Do they have multiple solutions? If yes, can you find more than one? Specify the domain over which the solutions exist. (a) ẏ = yt 2 (b) y = x 2 y 2 4x 2 (c) y = y+xe 2y/x x (d) 3tẏ = y cos(t), y() = 0 (e) ẏ = 3y 2/3, y(0) = 0 (f) yẏ + ( + y 2 ) sin(t) = 0, y(0) = 6. Solve dy dt = f ( ) at + by + m ct + dy + n where a, b,c, d,m, n are real constants. (Caveats: Do you need to assume any relation between these constants to solve this equation? What happens when this condition is not satisfied? On what domain do the solution(s) exist?) 7. (Ref. Section 3, pp. 92-98, Chap. 5 of Coddington) An equation is called exact if there exist an function F(x, y) such that M(x, y) + N(x, y)y = 0 (3) F x = M Prove Theorem 2. of page 93 of Coddington. (2) F y = N (4)
ODE Homework 2 Due Mon. 07 Sept. 2009; At the beginning of the class. Direct proof of uniqueness: Consider the same assumptions as Theorem 2.2 of page 0 of Coddington and Levinson (with the same notation). Let r(t) = ϕ (t) ϕ 2 (t). Following steps similar to those on page 9, prove that r(t) = 0 and use it to conclude that ϕ (t) = ϕ 2 (t). 2. Problems,2,3 on pages 37-38 of Coddington and Levinson. 3. Solving iterative inequalities for functions: (a) Consider the inequalities 0 (t) Mt and k (t) c First, show that if the equalities are valid in the above, then t 0 k (t) = M c k (s) ds () (ct) k+ (k + )!. (2) Use this fact to prove the following bound on k (t) in terms of M,c alone: (b) Consider the inequalities k (t) M c (ct) k+ (k + )!. (3) x n γδ n x 2 n and x h. (4) Find a bound on x n in terms of (γ,h, δ). Hint: First solve the equalities, to get x n as a function of (γ, h, δ). You could consider the following inequalities first, in order to practice finding the answer to this problem. x n x 2 n and x h. (5) x n γx 2 n and x h. (6) 4. Suppose the characteristic polynomial of a second order differential equation with constant coefficient has two complex solutions r,2 = a ± iω. Find two real linearly independent solutions. In general, if the characteristic polynomial of an n-th order equation has imaginary roots, is it possible to find real solutions of the corresponding n-th order equation? How? 5. Let y (t),...,y n (t) be n solutions of a general n-th order linear homogeneous equation y (n) + a (t)y (n ) + + p n (t)y = 0 (7) Show the Abel s formula for their Wronskian: W(y,...,y n )(t) = W(t 0 ) exp [ t ] p (s)ds t 0 (8)
6. Prove that the smooth solutions ϕ i (t), i =,...,k of the above equation, defined over an interval [a, b] are linearly independent iff their Wronskian is nonzero over that interval iff the Wronskian is non-zero at one point of the interval. 7. Let f(t, y) be a Lipschitz function (in y) over an interval I = [τ b, τ + b] with Lipschitz constant k. For a C -function x(t) defined over the interval, define its Picard iterate, which is another function over the same interval, by: T(x)(t) = ψ + t τ f(s, y(s))ds. (9) If we define the following norm of the function x(t) x = sup t I { e k t τ x(t) } (0) Show that T(y) T(x) α x y () for some α. 8. Forced vibrations: Consider the important equation for the position x(t) of a mass attached to a damped spring, which is forced by an external force. mẍ + γẋ + kx = F 0 cos(ωt), (2) where all the constants m, γ, k, F 0, ω are positive. Verify by a direct calculation that x = c e rt + c 2 e r2t + R cos(ωt delta) (3) solved the equation, where r r 2 are solutions of the characteristic polynomial, and R = F 0 sin(δ) = γω = m 2 (ω 2 0 ω2 ) 2 + γ 2 ω 2 (4) cos(δ) = m(ω2 0 ω 2 ) and ω 0 = k/m is the natural frequency of the unforced oscillation (which is the frequency of the time-periodic solution of the equation with F 0 = 0 = γ). Show that the first two terms decay, i.e., (5) u(t) U(t) = R cos(ωt δ) as t. (6) There is a whole lot to understand from this equation about forced oscillations, but we can spend some time in studying such applications in some extra classes. 2
ODE Homework 3 Due Fri. 09 Oct. 2009; At the beginning of the class This homework will be mainly concerned with describing methods for solving 2nd order ODE (with nonconstant coefficient). The main method for obtaining solutions (apart from the good old way of guessing a solution with enough practice) is in terms of power series.. Consider two functions a(x) and b(x) which have convergent power series for x < r a and x < r b, respectively: a(x) = α k x k and a(x) = β k x k () k=0 Let c(x) = a(x)b(x) and d(x) = a(x)/b(x) and f(x) = a(b(x)). Find the radius of convergence and the coefficients of power series of the functions c, d,f. 2. State comparison test and the Cauchy ratio test for the convergence of the power series. 3. Find the power series and radius of convergence for the following functions around x = 0. (A useful theorem about radius of convergence: Consider power series of a function f(x) about x = x 0. Let the independent variable x assume complex values. Let z 0 be the point in the complex plane where the f or one of its derivatives fails to exist. If x 0 z 0 = r 0, then the radius of convergence of the power series of f around x 0 is r 0.) (a) f(x) = x 2 (b) f(x) = +x 2 4. Problem 8 about Hermite polynomials of page 3 of Coddington. 5. Referring to Section 9 of Chapter 3 of Coddington (explaining the power series method), prove by induction the Equation (9.3) of page 4. 6. Problems 4. and 6.(a,b,c) of pages 49-50 of Coddington. 7. Find solutions of the Euler equation with initial values: k=0 x 2 y + axy + by = 0, y() = α, y () = β. Be sure to take care of the various cases (when the solutions to indicial polynomial are repeated or not, zero or not, etc.) 8. Problems. and 3. of page 54 of Coddington.
ODE Some practice problems on systems of equations These are similar to problems I have done at least partially in class. Of course, there were also many calculations that I did not finish completely and indicated that you should check them yourself. These will also be some of the practice problems.. Problem 5 of Section.5 of Perko. 2. Problem 9 of Section.5 of Perko. In addition, transform the solution back to the x, x 2 coordinates and thus write the general solution of that system. Is there any other method of solving this system and does it give the same general solution? 3. Is the following statement true? A matrix is nilpotent iff zero is the only eigenvalue. 4. Find the exponential exp(bt) for the matrix D I 0... 0 0 D I... 0 B =....... 0... D I 0... 0 D (a) First case when there are complex eigenvalues a ± ιb [ ] a b D = I = b a (b) Second case when there are real eigenvalues λ D = [ λ ] I = [ ] [ ] 0 0 5. Problems 2 and 3 of Section.7 of Perko (find the S + N decomposition of these matrices first). 6. Problems 6 of Section.8 of Perko (about finding Jordan form), and find the solution for the linear system with matrices in this problem. 7. For the matrices above, what are the stable, centre, and unstable subspaces E s, E c, and E u? For the two and three dimensional cases, sketch the vector field and the flow. 8. Problem 5 of Section.9 (about x 0 various subspaces) and related to it is Problem 2 of Section.8 of Perko. 9. Problems 2, 4, and 5 of Section 2.7 of Perko. Problems about stable/unstable manifolds I mentioned in the class (e.g. showing that a certain curve is a global stable and unstable manifold; finding it s local approximation using the steps in the proof of the stable manifold theorem, etc.) 0. Examples of Liapunov functions (from Section 2.9 of Perko).
ODE Homework Due Fri. 3 August 2009; At the beginning of the class. (a) Solve ẋ = x x 2 with initial condition x(0) = x 0. (b) Consider x 0 [0, ). Does the solution exist for all t [0, )? (c) Consider x 0 (,0). Does the solution exist for all t [0, )? (d) Does the solution exist for all t (,0] for each of the above cases? (e) Is there a T > 0 (possibly T = ), independent of x 0, such that the solution exists for all t [0,T] (or for t [0, ) if T = )? 2. Problems from Simmons: (a) # 4 on page 50 (b) # 0 on page 50 (c) # 2 on page 54 (d) # on page 59 (e) # 3(b) on page 62 (f) # 6 on page 62 (g) # 52 on page 77 3. Problems from Braun: (a) # 9 on page 0. Is this statement true if we simply assume a(t) > 0, instead of a(t) c > 0. (b) # 6 on page 0. Draw the vector field and the integral curves in the phase plane ((x, t)-plane) for this problem. 4. There are 50 differential equations on pages 75-76 of Simmons (in addition to many more in the exercises at the end of individual sections). If you solve all, or most, of them, that should be plenty to practice for solving st order differential equations! You should also practice drawing the vector fields and integral curves for the equations you solve. We will discuss some interesting ones in one of the discussion sessions on Fridays.
ODE Homework 2 Due Fri. 27 August 2009; At the beginning of the class. State the non-autonomous initial value problem (IVP) on R n, i.e., a system of n first order DE. E.g. refer to Coddington and Levinson book or the lecture notes. Show explicitly that it is equivalent to an autonomous IVP on R n+. (Part of the problem is to make precise what you mean by equivalent as per your definition of the IVP and then prove the equivalence above.) How about the above questions for IVP stated on C n, instead of R n? 2. Can you give a geometrical interpretation of the above? 3. Consider the equation dy dt = t+ye 2y/t t y(t 0 ) = y 0. () Convert this to an autonomous IVP. Can you solve the original equation? Can you solve the autonomous IVP? What restrictions need to be put on t 0 and/or y 0? 4. Consider a separable ODE (along with an initial value) as a non-autonomous IVP on R. What is the corresponding autonomous ODE on R 2? How do you relate the solution on R to that on R 2? How about the converse when is an autonomous ODE on R 2 equivalent to a separable ODE on R? 5. The same questions as in problem 4, but for an exact ODE.
ODE Homework 3 Due Wed. 06 October 200; At the beginning of the class. Problems, 2, and 3; pages 37-38 of Coddington-Levinson 2. Let g : R R be Lipschitz and f : R R continuous. Show that the system ẋ = g(x), ẏ = f(x)y, y(τ) = y 0, x(τ) = x 0, () has at most one solution on any interval containing τ. 3. Problems 9, 5, 6; page 80 of Braun 4. Show that for any ǫ > 0, there is an N such that the for all n > N, the n-th Picard iterate ϕ n is an ǫ-approximate solution. (I think this should be possible!! If you think this is implausible, state why you think so.) 5. Convert an n-th order linear differential equation into a system of ODE on R n of the form (x) = f(t,x) for x R n. Write the explicit form of the function f. Is that a Lipschitz continuous function (for all t)? Does it satisfy the conditions for existence and uniqueness of solutions of this system? (Justify your answers.) Can you use the theorem 2. of chapter of Coddington-Levinson (the difficult book) to prove Theorem 3, chapter 2, and theorem 2, chapter 3 of Coddington (the easy book). 6. Problem 2; page 80 of Braun. Furthermore, does there exist β > 0 such that the solution exists for t [0,T +β)? (Here, T is that constant involving 3 in the problem statement in Braun.) In addition, find the maximum T such that the solution exists for 0 t T for the same problem with the initial condition y(0) = y 0. (Clearly, T will now depend on y 0. Or will it?) In addition, state everything you can about this initial value problem. E.g. what is the interval of existence and uniqueness if you change the initial time and initial condition? Now T will be a function of both t 0 and y 0 (maybe??). ẏ = e (y t)2 y(t 0 ) = y 0. (2) What happens to a solution of this problem for y 0 < 0 and t? What happens to a solution of this problem for y 0 > 0 and t? Can you find a solution for t 0 = 0 and y 0 = 0? Is it unique? Its interval of existence? 7. The same as above problem, but for the equation Please see next page as well. ẏ = e (y t) (3)
A note about use of matlab to solve ODE. It s quite simple really. Here are the steps:. Open any text editor, like emacs or vi and create a file called fn.m with the following contents function dy = fn(t,y) dy = exp((y-t)^2) end Of course, this is for solving the equation from problem 6. Replace the RHS by appropriate f(t,y) for other equations. 2. To solve the ODE from time t0 to t with y(t0) = y0 and then plot it, use the following commands in matlab [tsol ysol] = ode45(@fn, [t0 t], [y0]); plot(tsol, ysol); E.g. you can do the following: t0 = 0; t = 0.; y0 = ; [tsol ysol] = ode45(@fn, [t0 t], [y0]); plot(tsol, ysol); It is allowed to have t < t0, e.g., t0 = 0.5;t = 0.3 etc. Type in matlab help ode45 or search the web for more help (or stop by my office). This will be useful for a really complete answer to problem 6! 2