Liouville-Green exponential ansatz

Writing $v=e^S$ in $v^{\prime \prime }=q(x)v$ produces the Riccati equationExpanding $S^{\prime }$ in inverse powers of $x$ determines the exponential phase and algebraic prefactor recursively.

A Riccati equation is a first-order equation quadratic in the unknown function.

For constant nonzero $c$, putting $y=x^{\prime }/(cx)$ in $y^{\prime }+c{y}^2+a(t)y+b(t)=0$ cancels the quadratic derivative terms and gives $x^{\prime \prime }+a(t)x^{\prime }+cb(t)x=0$. The logarithmic derivative is defined on intervals where $x\ne 0$, and multiplying $x$ by a nonzero scalar does not change $y$. For variable $c(t)\ne 0$, the derivative coefficient becomes $a-c^{\prime }/c$.

At a regular point of a second-order linear ordinary differential equation, a nontrivial solution cannot have both $x$ and $x^{\prime }$ zero, by initial-value uniqueness. Its zeros are therefore simple. Under the linearization of a Riccati equation, such a zero $t_{\ast }$ produces $y=x^{\prime }/(cx)\sim 1/[c(t-t_{\ast })]$. Zeros separate the intervals on which the Riccati equation solution is finite.