Page 18 - 6637

Supplementary Problems

√
(g) u( 10, 1) = 5.

Exercise 43. Solve the equation xu + (x + y)u = 1 with the initial conditions
x
y
u(1, y) = y. Is the solution defined everywhere?

Exercise 44. Let p be a real number. Consider the PDEs xu + yu = pu,
y
x
−∞ < x < ∞, −∞ < y < ∞.
(a) Find the characteristic curves for the equations.
2
2
(b) Let p = 4. Find an explicit solution that satisfies u = 1 on the circle x +y = 1.
2
(c) Let p = 2. Find two solutions that satisfy u(x, 0) = x , for every x > 0.
2
Exercise 45. The equation xu + (x + y)u + (y/x − x)u = 1 is given along
x
y
with the initial condition u(1, y) = 0.
(a) Solve the problem for x > 0. Compute u(3, 6).

(b) Is the solution defined for the entire ray x > 0?

2
Exercise 46. Solve the Cauchy problem u + u = u , u(x, 0) = 1.
x
y
Solution: The parametric solution is (x(t, s), y(t, s), u(t, s)) = (t + s, t, 1 ), im-
1−t
plying u = 1/(1 − y).
2
Exercise 47. Solve the equation xuu + yuu = u − 1 for the ray x > 0 under
x
y
2
3
the initial condition u(x, x ) = x .
Exercise 48. A river is defined by the domain D = {(x, y): |y| < 1, −∞ <
x < ∞}. A factory spills a contaminant into the river. The contaminant is further
spread and convected by the flow in the river. The velocity field of the fluid in the
river is only in the x direction. The concentration of the contaminant at a point

(x, y) in the river and at time τ is denoted by u(x, y, τ). Conservation of matter
2
and momentum implies that u satisfies the first-order PDE u − (y − 1)u = 0.
τ
x
y −x
The initial condition is u(x, y, 0) = e e 2 .
(a) Find the concentration u for all (x, y, τ).
(b) A fish lives near the point (x, y) = (2, 0) at the river. The fish can tolerate
contaminant concentration levels up to 0.5. If the concentration exceeds this
level, the fish will die at once. Will the fish survive? If yes, explain why. If
no, find the time in which the fish will die. Hint: Notice that y appears in the

PDE just as a parameter.

2
Exercise 49. Solve the equation (y + u)u + yu = 0 in the domain y > 0,
x
y
2
under the initial condition u = 0 on the planar curve x = y /2.
2
Exercise 50. Find a function u(x, y) that solves the Cauchy problem (x+y )u +
x
x
yu + ( − y)u = 1, u(x, 1) = 0, x ∈ R.
y
y

14