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Solved Problems

(b) φ xx + xφ yy = 0

The equation in part (b) is known as Tricomi’s equation and is a model for transonic

fluid flow in which the flow speed changes from supersonic to subsonic.
Solution:

(a) For y 6= 0, the equation is hyperbolic. We find the new independent variables.

p
dy y 2 x −x −x
= = y, y = c e , e y = c, ξ = e y
dx 1
p
dy − y 2
x
x
−x
= = −y, y = c e , e y = c, ψ = e y
dx 1
Next we determine x and y in terms of ξ and ψ.
2
ξψ = y , y = p ξψ

1 ψ
x
p
p
ψ = e x ξψ, e = ψ/ξ, x = log
2 ξ
We calculate the derivatives of ξ and ψ.
ξ = − e −x y = −ξ
x
ξ = e −x = p ξ/ψ
y
x
ψ = e y = ψ
x
x
ψ = e = p ψ/ξ
y
Then we calculate the derivatives of φ.
s s
∂ ∂ ∂ ∂ ξ ∂ ψ ∂
= −ξ + ψ , = +
∂x ∂ξ ∂ψ ∂y ψ ∂ξ ξ ∂ψ
s s
ξ ψ
φ = −ξφ + ψφ , φ = φ + φ ψ
ψ
x
y
ξ
ξ
ψ ξ
ξ ψ
2
2
φ xx = ξ φ − 2ξψφ ξψ + ψ φ ψψ + ξφ + ψφ , φ yy = φ + 2φ ξψ + φ ψψ
ξ
ξξ
ψ
ξξ
ψ ξ
Finally we transform the equation to canonical form.
2
2
φ xx − y φ + φ − φ + x = 0
x
yy

ψ
−4ξψφ ξψ + ξφ + ψφ − ξφ + ψφ − φ + log = 0
ψ
ξ
ξ
ψ
ξ

1 ψ
φ ξψ = φ + φ − log
ψ
2ξ ξ
For y = 0 we have the ordinary differential equation
2
φ xx + φ − φ + x = 0.
x
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