Question

The pde uxx+ x^2 uxy -(x^2/2+1/4) uyy = 0 is hyperbolic in the entire xy-plane.true or false why?

Answer 1

Answer on Question #66108 - Math – Differential Equations

Question

The pde uxx + x^2 uxy - (x^2/2 + 1/4) uyy = 0 is hyperbolic in the entire xy-plane. true or false, why?

Solution

We have

$$u_{xx} + x^2 u_{xy} - \left(\frac{x^2}{2} + \frac{1}{4}\right) u_{yy} = 0$$

which is a particular case of the linear second-order partial differential equation

$$A u_{xx} + B u_{xy} + C u_{yy} + D u_x + E u_y + F = 0$$

where coefficients $A, B, C, D, E$ and free term $F$ in general are functions of the independent variables $x, y$, but do not depend on the unknown function $u$.

This equation is said to be hyperbolic if [1, page 435]

$$B^2 - 4AC > 0$$

For the given equation (1)

$$A = 1, \quad B = x^2, \quad C = - \left(\frac{x^2}{2} + \frac{1}{4}\right), \quad D = E = F = 0$$

Substituting values $A, B, C$ in (2) we get

$$B^2 - 4AC = x^4 + 4 \left(\frac{x^2}{2} + \frac{1}{4}\right) = x^4 + 2x^2 + 1 = (x^2 + 1)^2 > 0$$

This inequality holds for any $x$. Thus, this equation is hyperbolic in the entire xy -plane.

**Answer**: True. The pde $u_{xx} + x^2 u_{xy} - \left(\frac{x^2}{2} + \frac{1}{4}\right) u_{yy} = 0$ is hyperbolic in the entire xy-plane.

References:

[1] Dennis G. Zill, Michael R. Cullen. Differential equations, seventh edition.

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