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Supplementary Problems

Exercise 35. Solve au +bu = f(x, y), where f(x, y) is a given function. Write
y
x
the solution in the form
Z
2 1/2
2
u(x, y) − (a + b ) fds + g(bx − ay),
L
where g is an arbitrary function of one variable, L is the characteristic line segment
from the y axis to the point (x, y), and the integral is a line integral. (Hint: Use
the coordinate method.)

Exercise 36. Use the coordinate method to solve the equation u + 2u + (2x −
y
x
2
2
y)u = 2x + 3xy − 2y .
Exercise 37. Find the most general solutions u(x, y) of the following equations
consistent with the boundary conditions stated:
(a) y ∂u − x ∂u = 0, u(x, 0) = 1 + sin x;
∂x ∂y
∂u
2
(b) i ∂u = 3 , u = (4 + 3i)x on the line x = y;
∂x ∂y
(c) sin x sin y ∂u + cos x cos y ∂u = 0, u = cos 2y on x + y = π/2;
∂x ∂y
2
(d) ∂u + 2x ∂u = 0, u = 2 on the parabola y = x .
∂x ∂y
Exercise 38. Find solutions of 1 ∂u + 1 ∂u = 0 for which (a) u(0, y) = y, (b)
x ∂x y ∂y
u(1, 1) = 1.

Exercise 39. Find the most general solutions u(x, y) of the following equations

consistent with the boundary conditions stated:
2
(a) y ∂u − x ∂u = 3x, u = x on the line y = 0;
∂x ∂y
(b) y ∂u − x ∂u = 3x, u(1, 0) = 2;
∂x ∂y
3
3
2 2
(c) y 2 ∂u + x 2 ∂u = x y (x + y ), no boundary conditions.
∂x ∂y
Exercise 40. Solve sin x ∂u + cos x ∂u = cos x subject to (a) u(π/2, y) = 0, (b)
∂x ∂y
u(π/2, y) = y(y + 1).

Exercise 41. A function u(x, y) satisfies 2 ∂u + 3 ∂u = 10, and takes the value 3
∂x ∂y
on the line y = 4x. Evaluate u(2, 4).

Exercise 42. In those cases in which it is possible to do so, evaluate u(2, 2),
2
2
where u(x, y) is the solution of 2y ∂u − x ∂u = 2xy(2y − x ) that satisfies the
∂x ∂y
(separate) boundary conditions given below.
2
(a) u(x, 1) = x for all x.
2
(b) u(x, 1) = x for x ≥ 0.
2
(c) u(x, 1) = x for 0 ≤ x ≤ 3.
(d) u(x, 0) = x for x ≥ 0.

(e) u(x, 0) = x for all x.
√
(f) u(1, 10) = 5.

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