Question

Which of the following statements is true?

(a) Hyperbolic equations have three families of characteristic curves

(b) Hyperbolic equations have one family of characteristic curves

(c) Hyperbolic equations have no families of characteristic curves

(d) Hyperbolic equations have two families of characteristic curves

This question was addressed to me in exam.

I'd like to ask this question from Solution of Second Order P.D.E. in portion Partial Differential Equations of Engineering Mathematics

Answer 1

The correct option is (d) Hyperbolic equations have two families of characteristic curves

To explain: The canonical variables ξ and η for a hyperbolic pde satisfy the equations,

\(aζ_x+(b+\sqrt{b^2-ac}) ζ_y=0 \, and \, aη_x+(b+\sqrt{b^2-ac}) η_y=0\)

The families of curves ξ = constant and η = constant are the characteristic curves. Hence, hyperbolic equations have two families of characteristic curves.