Is there any way to solve it without numerical way??
$ \frac{d^2 y}{d x^2}= \frac{1}{y}$ thanks in advance!!
Is there any way to solve it without numerical way??
$ \frac{d^2 y}{d x^2}= \frac{1}{y}$ thanks in advance!!
Note that $y''y'=y'/y$ hence $(y')^2=c+2\log|y|$ hence $y'=\pm\sqrt{c+2\log|y|}$ and $ \int_{y(0)}^{y(x)}\frac{\mathrm dt}{\sqrt{c+2\log|t|}}=\pm x. $ The LHS does not seem to be (the inverse of) a usual function of $y(0)$ and $y(x)$. An equivalent formulation is $ \mathrm e^{-c/2}\int_{\sqrt{c+2\log|y(0)|}}^{\sqrt{c+2\log|y(x)|}}\mathrm e^{t^2/2}\mathrm dt=\pm x, $ and the LHS can be rewritten using the imaginary error function $\mathrm{erfi}$, with no obvious gain.