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u y = 2yu σ
2
u xx = 4x u ττ + 2u τ
2
u yy = 4y u σσ + 2u σ
Finally we transform the equation to canonical form.

2
2
y u xx + x u yy = 0
σ(4τu ττ + 2u τ ) + τ(4σu σσ + 2u σ ) = 0
1 1
u σσ + u ττ = − u σ − u τ
2σ 2τ

Conclusion 3.1 The second order linear PDE’s can be classified into three types,
which are invariant under changes of variables. The types are determined by the sign
of the discriminant. This exactly corresponds to the different cases for the quadratic
equation satisfied by the slope of the characteristic curves. We saw that hyperbolic
equations have two distinct families of (real) characteristic curves, parabolic equations
have a single family of characteristic curves, and the elliptic equations have none. All
the three types of equations can be reduced to canonical forms. Hyperbolic equations
reduce to a form coinciding with the wave equation in the leading terms, the parabolic
equations reduce to a form modeled by the heat equation, and the Laplace’s equation
models the canonical form of elliptic equations. Thus, the wave, heat and Laplace’s
equations serve as canonical models for all second order constant coefficient PDE’s.

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