By David Bleecker, George Csordas (auth.)

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Solve P'(t) = bP(t) - ct in terms of b, c, P(O) and t. Under what condition(s) (on P(O), b and c) will P(t) drop to zero for some t > 0 ? 6. Find the general solution, y(x), of the following second-order homogeneous linear ODEs. (a) y" = 0 (d) y" + y' = 0 (b) y" - 3y = 0 (c) y" + 3y = 0 (e) y" -3y' = 0 (f) 4y" + 3y' + 5y = 0 (g) 2y" + 5y' + 2y = 0 (h) y" -6y' + 13y = 0 (i) y"-4y' +4y=0 (j) y" + 10 y' + 25y = 0 . 7. Find the particular solutions y( t), meeting the given initial data, of the following second-order homogeneous linear ODEs.

Assume that the orbit is not a perfect circle. (i) If n = 4, show that either r(t) -+ 00 as t -+ 00, or r(t) -+ 0 as t approaches some finite value. (ii) If n ~ 5, show that in addition to the tW0 possibilities in (i), it can also rarely happen that the orbit will spiral toward a circular orbit. Show that this can only occur if the maximum value of f(r) is E. Why is this a rare occurance ? 2 Generalities about PDEs 43 20. 3255· R. (a) For C > 0, consider the curve r = C·cosh(z/C) in the zr-plane.

For example, if f(x) = sin(7IX) and g(x) = sin(371X), the theory of Chapter 5 yields the solution u(x,t) = cos( 1I'at)sin( 7IX) + (1/31I'a)sin(311'at)sin(371X). (14) 0 Linear PDEs, Classification, and the Superposition Principle All of the PDEs in the above examples are linear. The notion of linearity for PDEs is strictly analogous to linearity for ODEs. Recall that the general n-th order linear ODE is an ODE which is expressible in the form ( 15) where ao(x), a1(x), ... , an(x) and f(x) are given (possibly constant) functions of the independent variable x.