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Supplementary Problems

φ = −φ − iφ β
ψ
α
φ ξψ = −φ αα − φ ββ

We transform the equation to canonical form.

φ + φ ψ
ξ
φ ξψ = −
12(ξ + ψ)
−2iφ β
−φ αα − φ ββ = −
12iβ

φ β
φ αα + φ ββ = −
6β

3.2 Supplementary Problems

Exercise 68. Determine the type of the PDEs. By means of a method of char-

acteristics bring PDEs to its canonical view:
2
2
2
2
a) x u xx − y u yy = 0; b) sin xu xx − 2y sin xu xy + y u yy = 0;
c) u xx − 2u xy + 2u yy = 0.
2
3
Exercise 69. What is the order of the following equations (a) ∂ u + ∂ u = 0,
∂x 3 ∂y 2
4
2
2
(b) ∂ u − 2 ∂ u + ∂ u = 0?
2
∂x 2 ∂x ∂y ∂y 2
2
Exercise 70. Classify the following differential equations (as elliptic, etc.) (a) ∂ u −
∂x 2
2
2
2
2
2
2
2
2
2 ∂ u + ∂ u = 0, (b) ∂ u + ∂ u + ∂u = 0, (c) ∂ u − ∂ u +2 ∂u = 0, (d) y ∂ u +x ∂ u = 0.
∂x∂y ∂y 2 ∂x 2 ∂y 2 ∂x ∂x 2 ∂y 2 ∂y ∂x 2 ∂y 2
Exercise 71. What are the types of the following equations? (a) u xx − u xy +
2u + u − 3u yx + 4u = 0. (b) 9u xx + 6u xy + u + u = 0.
yy
yy
x
y
Exercise 72. Find the regions in the xy-plane where the equation (1 + x)u xx +
2
2xyu xy − y u yy = 0 is elliptic, hyperbolic, or parabolic. Sketch them.
Exercise 73. What is the type of the equation u xx − 4u + 4u yy = 0? Show by
xy
direct substitution that u(x, y) = f(y + 2x) + xg(y + 2x) is a solution for arbitrary
functions f and g.
Exercise 74. Reduce the elliptic equation u xx + 3u − 2u + 24u + 5u = 0 to
x
yy
y
the form v xx + v yy + cv = 0 by a change of dependent variable u = ve αx+βy and
0
then a change of scale y = γy.
Exercise 75. Consider the equation 3u +u xy = 0. (a) What is its type? (b) Find
y
the general solution (Hint: Substitute v = u ) (c) With the auxiliary conditions
y
u(x, 0) = e −3x and u (x, 0) = 0, does a solution exist? Is it unique?
y
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