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Supplementary Problems

Exercise 76. For what values of the constant k is u xx + ku + u yy = 0 elliptic?
xy
Parabolic? Hyperbolic?

Exercise 77. In what regions of the xy plane is (1+y)u +2xu +(1−y)u yy =
xx
xy
u elliptic? Parabolic? Hyperbolic?
x
2
Exercise 78. Consider the equation u xx − 6u xy + 9u yy = xy .
(a) Find a coordinates system (s, t) in which the equation has the form: 9v =
tt
2
13(s − t)t .
(b) Find the general solution u(x, y).
(c) Find a solution of the equation which satisfies the initial conditions u(x, 0) =
sin x, u (x, 0) = cos x for all x ∈ R.
y

Exercise 79.
(a) Show that the following equation is hyperbolic: u xx + 6u xy − 16u yy = 0.
(b) Find the canonical form of the equation.

(c) Find the general solution u(x, y).
(d) Find a solution u(x, y) that satisfies u(−x, 2x) = x and u(x, 0) = sin 2x.

Exercise 80. Consider the equation u xx + 4u xy + u = 0.
x
(a) Bring the equation to a canonical form.
(b) Find the general solution u(x, y) and check by substituting back into the
equation that your solution is indeed correct.
(c) Find a specific solution satisfying u(x, 8x) = 0, u (x, 8x) = 4e −2x .
x

5
Exercise 81. Consider the equation y u xx − yu + 2u = 0, y > 0.
yy
y
(a) Find the canonical form of the equation.
(b) Find the general solution u(x, y) of the equation.
3
(c) Find the solution u(x, y) which satisfies u(0, y) = 8y , and u (0, y) = 6, for
x
all y > 0.

2
2 2
Exercise 82. Consider the equation u xx + (1 + y ) u − 2y(1 + y )u = 0.
yy
y
(a) Find the canonical form of the equation.
(b) Find the general solution u(x, y) of the equation.
(c) Find the solution u(x, y) which satisfies u(x, 0) = g(x), and u (x, 0) = f(x),
y
2
where f, g ∈ C (R).
(d) Find the solution u(x, y) for f(x) = −2x, and g(x) = x.

Exercise 83. Consider the equation u +2u +[1−q(y)]u yy = 0, where q(y) =
xy
xx

−1, y < −1,

0, |y| ≤ 1,

1, y > 1.

(a) Find the domains where the equation is hyperbolic, parabolic, and elliptic.

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