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Supplementary Problems

(b) For each of the above three domains, find the corresponding canonical trans-
formation and the canonical form.

(c) Draw the characteristics for the hyperbolic case.

2y
2
2
Exercise 84. Consider the equation 4y u +2(1−y )u −u − 1+y 2 (2u −u ) =
xx
x
y
xy
yy
0.
(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) are arbitrary functions.
y
Exercise 85. Consider the equation u xx − 2u xy + 4e = 0.
(a) Find the canonical form of the equation.
(b) Find the solution u(x, y) which satisfies u(0, y) = f(y), and u (0, y) = g(y).
x

Exercise 86. Consider the equation u xx + yu yy = 0. Find the canonical forms of
the equation for the domain where the equation is hyperbolic, and for the domain
where it is elliptic.

Exercise 87. State the nature of each of the following equations (that is, whether
2
2
2
2
elliptic, parabolic, or hyperbolic) (a) ∂ y + α ∂ y = 0; (b) x ∂ u + y ∂ u + 3y 2 ∂u = 0.
∂t 2 ∂x 2 ∂x 2 ∂y 2 ∂x

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