Logarithmic-derivative reduction of a second-order equation ($y^{\prime \prime }=(y^{\prime }{)}^2/y+a(x)y^{\prime }⟹(y^{\prime }/y{)}^{\prime }=a(x)(y^{\prime }/y)$)

Where $y\ne 0$, its logarithmic derivative $q=y^{\prime }/y$ changes this nonlinear second-order ordinary differential equation into the first-order linear differential equation $q^{\prime }=a(x)q$. For continuous $a$, initial values $y(x_0)=y_0\ne 0$ and $y^{\prime }(x_0)=v_0$ therefore giveDifferentiation proves both the equation and the initial data. The exponential preserves the sign of $y_0$, so the solution never reaches the singular value zero on any finite interval where $a$ is continuous. For $a(x)=2x$ and $y_0=v_0=1$, the expression reduces to $y(x)=\exp ({\int }_0^x{e}^{t^2}dt)$.