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CHAPTER 42
42.55 Show that f(x, t) = (x + at)3 satisfies the wave equation, a2ffx=fn.
fx=3(x + at)2, fxx = 6(x + at). f, = 3(x + at)2(a) = 3a(x + at)2, /„ = 6a(x + at)(a) = 6a2(* + at) = a2/,,.
42.56 Show that f(x, t) = sin (x + at) satisfies the wave equation a2fxll=fll.
fx = cos (A: + at), fxx = —sin (x + at), f, = a cos (x + at), /„ = — a2 sin (x + at) = a2fxx.
42.57 Show that f(x,t) = e* "' satisfies the wave equation a2fxx=ftt.
/, = «*"". /„ = «*"" /, = -«"", /,, = flV-'= «*/„.
42.58 Let /(jt, f) = M(X + af) + v(x — at), where u and v are assumed to have continuous second partial derivatives.
In generalization of Problems 42.55-42.57, show that / satisfies the wave equation a2fxx =/„.
fx = u'(x + at) + v'(x-at), fxx = u"(x + at) + v"(x-at). f, = au'(x + at) - av'(x - at), fl, = a2u"(x +
at) + a2v"(x - at) = a2fxjc.
42.59
Verify the general formula
f(x, y)dy =
dy for f(x, y) = sin xy, a = 0, b = TT.
can be integrated by parts:
On the other hand,
So,
and
42.60 Verify the general formula
for f(x, y) = x + y, a = 0, b = l.
Hence,
On the other hand,
42.61
If f(x, y) = /„' cos (x + 2y + t) dt, find / v and f y .
fy = x2 + yx,
Ly=1x + y,
42.63 Let M(x, y) and N(x, y) satisfy dMldy = 9Nldx for all (x, y). Show the existence of a function /(x, y) such
that dfldx = M and dfldy = N.
Let
Then
Also,
since
42.62 Let f(x, y) = J (x2 + tx) dt. Find fx and fy and verify that fxy=fyjl.