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Supplementary Problems

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Exercise 51. Solve the equation u + u u = 0 in the ray x > 0 under the initial
x
y
√
condition u(x, 0) = x.
Exercise 52. Consider the equation uu + xu = 1, with the initial condition
y
x
1 2
1 3
( s + 1, s + s, s). Find a solution.
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Exercise 53. Consider the PDE xu + yu = 1/ cos u. Find a solution to the
x
y
2
equation that satisfies the condition u(s , sin s) = 0 (you can write down the
solution in the implicit form F(x, y, u) = 0).
1
Exercise 54. Consider the PDE uu + u = − u. Find a solution satisfying
y
x
2
2
u(x, 2x) = x .
Exercise 55. Solve the Cauchy problem yu − uu = x, u(s, s) = −2s, −∞ <
y
x
s < ∞.
2
Exercise 56. Solve the Cauchy problem u + u = 0, u(x, 0) = x.
y
x
2
Exercise 57. Let u(x, t) be the solution to the Cauchy problem u +cu +u = 0,
t
x
u(x, 0) = x, where c is a constant, t denotes time, and x denotes a space coordinate.
Solve the problem.

Exercise 58. Solve the problem xu − uu = y, u(1, y) = y, −∞ < y < ∞.
y
x
Exercise 59. Solve the problem xu − yu + u = 0, u(x, 0) = 1, x > 0.
x
y
Exercise 60. Consider the equation xu + (1 + y)u = x(1 + y) + xu. Find the
x
y
general solution.

Exercise 61. Determine the general solution of each of the following equations
(with a, b, and c constant), writing each solution in a form solved for z : (a) a ∂z +
∂x
b ∂z = c, (b) a ∂x + b ∂z = cz, (c) y ∂z − x ∂z = 0, (d) ∂z + ∂z + 2xz = 0,
∂y ∂x ∂y ∂x ∂y ∂x ∂y
2
(e) x ∂z − y ∂z = z, (f) x 2 ∂z + y 2 ∂z = z .
∂x ∂y ∂x ∂y

Exercise 62. Find the solution of the equation ∂z = ∂z for which z = (t + 1) 4
∂x ∂y
2
when x = t + 1 and y = 2t.

Exercise 63. Determine the general solution of each of the following equations,
writing each solution in a form solved for z : (a) (x + y)( ∂z + ∂z ) = z − 1, (b)
∂x ∂y
∂z = xy, (c) xz ∂z + yz ∂z = xy.
∂x ∂x ∂y

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