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

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