18.03 Problem Set 3

Due by 1:00 P.M., Friday, March 5, 1999, in the boxes at 2-106, next to the Undergraduate Mathematics Office.

Syllabus: Part II: Second order linear equations

9. (M 22 Feb) Second order linear equations: EP 2.1, 2.2.

10. (W 24 Feb) Complex numbers: Notes C.

11. (F 26 Feb) Complex-valued functions; the case of real roots: EP 2.3.

12. (M 1 Mar) Non-real and repeated roots: 2.3, 2.4.

13. (W 3 Mar) Initial value problems, Wronskian, operators: Notes O 4–8.

14. (F 5 Mar) Operators; higher order equations: Notes O 4–8.

Part I.

12. (M 1 Mar) EP 2.1: 39, 41. A spring weakens very slowly over time: the spring constant k decreases, but so slowly that the motion at any given moment is as if it were constant. Assume that we have a system with this spring, a dashpot with constant c, and a mass m, controlled by the ODE

mẍ + cẋ + kx = 0.

At what value of k does the system stop exhibiting “underdamped” oscillatory behavior and transition to “overdamped” exponential decay?

13. (W 3 Mar) In each of EP 2.1: 33, 35, 37, 39, 41, determine all solutions which do not become arbitrarily large in absolute value as x goes to infinity.

Part II.

12. (M 1 Mar) This problem involves the “Wronskian,” which is described in EP 2.1. Consider the homogeneous second order linear equation

y′′ + p(x)y′ + q(x)y = 0  (1)

defined on some interval, and let y_{1} and y_{2} be any two solutions to it. The “Wronskian” of these two solution is

w = y_{1}y′_{2} − y′_{1}y_{2}.

Notice that if we multiply y_{2} by a constant, then the Wronskian gets multiplied by the same constant.

(a) Show that w′ = y_{1}y_{2}′′ − y_{1}′′y_{2}.

(b) Show that w satisfies the homogeneous first order linear equation

w′ + p(x)w = 0.  (2)

This is very interesting, since (2) does not depend at all upon the specific solutions y_{1} and y_{2}. By separation of variables,

w = ce^{− ∫ p(x)dx}  (3)

for some constant c. This constant of integration c is the only dependence w has on the pair of solutions. (3) is Abel’s formula.

(c) Conclude that either w(x) is zero for all x, or else w(x) is zero for no x. Also: if p = 0, w is constant.

(d) For example, find a linearly independent pair of solutions of y′′ +ay = 0 (whose form will depend upon the constant a) and verify that the Wronskian is constant.

(e) Abel’s formula can sometimes be used to find a second solution to (1) if we have a first solution. For example consider

x^{2}y′′ + xy′ − y = 0  (4)

on the half line where x > 0. Verify that y_{1} = x is a solution. Use Abel’s formula (taking c = 1) to compute the Wronskian. Then use the definition of the Wronskian to write down an inhomogeneous first order linear differential equation satisfied by y_{2}. Finally, solve this equation and verify that its solution satisfies (4).

13. (W 3 Mar) Nothing new.