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Solutions to Exercises

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Exercise 33. By either the coordinate method u = e −c(ax+by)/(a +b ) f(bx − ay),
where f is arbitrary.

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Exercise 36. u = x + 2y + 5/(y − 2x) + exp[(−2x − 3xy + 2y )/5]f(y − 2x)
where f is arbitrary.

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Exercise 37. We have (a) p = x + y , u = sin(x + y ) + 1; (b) p = 3x + iy,
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u = (3x + iy) 1/2 /2; (c) p = sin x cos y, u = 2 sin x cos y − 1; (d) p = y − x ,
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u = y − x + 2.

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Exercise 38. (a) (y − x ) ; (b) 1 + f(y − x ) where f(0) = 0.

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Exercise 39. (a) p = x + y , particular integral u = −3y, u = x + y − 3y; (b)
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u = x + y2 − 3y + 1 + g(x + y ) where g(1) = 0; (c) (x + y )/6 + g(x − y ).

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Exercise 40. u = y+f(y−ln(sin x)); (a) u = ln(sin x); (b) u = y+[y−ln(sin x)] .

Exercise 41. u = f(3x − 2y) + 2(x + y); f(p) = 3 + 2p; u = 8x − 2y + 3 and
u(2, 4) = 11.

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Exercise 42. p = x + 2y ; u(x, y) = f(p) + x y /2.
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(a) u(x, y) = (x + 2y + x y − 2)/2. u(2, 2) = 13. The line y = 1 cuts each
characteristic in zero or two distinct points, but this causes no difficulty with
the given boundary conditions.
(b) As in 1.
(c) The solution is defined over the space between the ellipses p = 2 and p = 11;
(2, 2) lies on p = 12, and so u(2, 2) is undetermined.
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(d) u(x, y) = (x + 2y ) + x y /2; u(2, 2) = 8 + 12.
(e) The line y = 0, cuts each characteristic in two distinct points. No differentiable
form of f(p) gives f(±a) = ±a respectively, and so there is no solution.
(f) The solution is only specified on p = 21, and so u(2, 2) is undetermined.
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(g) The solution is specified on p = 12, and so u(2, 2) = 5 + (4)(4) = 13.
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Exercise 44. Below we present our solutions.
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(a) The parametric solution is x(t) = x e , y(t) = y e , u(t) = u e , and the
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characteristics are the curves x/y = constant.
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(b) u(x, y) = (x + y ) is the unique solution.
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