By David Bleecker, George Csordas (auth.)
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Extra resources for Basic Partial Differential Equations
B) Show that the graph of any solution u(x,y) of Laplace's equation of the form in (a), intersects the xy-plane in a pair of perpendicular lines through (0,0). 8. C. f(x) = sin(n11X), if and only if n is an integer. (b) In how many points does the graph of sin(n11X) intersect the x-axis between 0 and 1 ? (c) Give a physical reason for why the temperature approaches 0 faster if n is larger. 9. Let u(x,y,z,t) be the solution (11) in Example 3 on wave problems. '/ A) u, etc.. Use these facts to deduce that u(x,y,z,t) satisfies the wave equation (9).
C) Use the definition ~ [f(x) + ig(x)] e(a+i(J)x = f/(x) + ig/(x) and the formula = ell'Xei(Jx = ell'X[cos((Jx) + isin((Jx)] to show that ~ lX = rlx ,where r = a + i(J . (d) Use part (c) to verify that the function y(x) defined by (23) (see also (21) and (22)) satisfies the differential equation (18). 12. Find the constants c1 and c2 in (35) and (36) such that x(O) = Xo and x' (0) = 0 . 13. Suppose that in Example 7 (with LRC f. 0), the voltage source is alternating, say V(t) = sin(wt). For what value of w is the amplitude of i(t) the greatest, for large t ?
C. (10). (c) The set of points (x,y,z) inside the box, where u(x,y,z,t) is always zero, is the union of a number of intersecting rectangular surfaces which divide the interior of the box into a number of compartments. How many compartments are there? (d) At what points in the box does the pressure experience the greatest changes? 10. Refer to Example 3, and assume that the box is cubical with A =B= C =1 and a = 2. (a) By givine; an example, show that it is possible for two independent solutions of the form of u(x,y,z,t) in (11) to have the same frequency.
Basic Partial Differential Equations by David Bleecker, George Csordas (auth.)