On a second-order boundary value problem arising in combustion theory

We obtain existence and uniqueness results for the boundary-value problem \[ y = x 2 − y 2 , y ( x ) ∼ ∓ x a s x → ± ∞ y = {x^2} - {y^2}, \qquad y\left ( x \right ) \sim \mp x \qquad as \qquad x \to \pm \infty \] . Our main result shows that there are precisely two solutions y + ( x ) > − | x | {y_+} \left ( x \right ) > - \left | x \right | and y − ( x ) > − | x | {y_-}\left ( x \right ) > - \left | x \right | . Only the latter is of physical interest in the problem in combustion theory from which the equation arises.