Second Order ODE/y'' = f(y)

Theorem

$\dfrac {\d^2 y} {\d x^2} = \map f y$

$\dfrac {\sqrt 2} 2 \ds \int \dfrac {\d y} {\sqrt {\ds \int \map f y \rd y + C_1} } = x + C_2$

where $C_1$ and $C_2$ are arbitrary constants.

$\ds \dfrac {\d^2 y} {\d x^2}$

$=$

$\ds \map f y$

$\ds \leadsto \ \ $

$\ds 2 \dfrac {\d y} {\d x} \dfrac {\d^2 y} {\d x^2}$

$=$

$\ds 2 \map f y \dfrac {\d y} {\d x}$

$\ds \leadsto \ \ $

$\ds \int 2 \dfrac {\d y} {\d x} \dfrac {\d^2 y} {\d x^2} \rd x$

$=$

$\ds \int 2 \map f y \dfrac {\d y} {\d x} \rd x$

$\ds \leadsto \ \ $

$\ds \paren {\dfrac {\d y} {\d x} }^2$

$=$

$\ds 2 \int \map f y \rd y + C$

$\ds \leadsto \ \ $

$\ds \dfrac {\d y} {\d x}$

$=$

$\ds \sqrt {2 \int \map f y \rd y + C_1}$

$\ds \leadsto \ \ $

$\ds \int \dfrac {\d y} {\sqrt {2 \ds \int \map f y \rd y + C_1} }$

$=$

$\ds \int \rd x$

$\ds \leadsto \ \ $

$\ds \dfrac {\sqrt 2} 2 \int \dfrac {\d y} {\sqrt {\ds \int \map f y \rd y + C_1} }$

$=$

$\ds x + C_2$

$\blacksquare$

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation: differential equations of the second order: $(2)$ Equations of the form $\dfrac {\d^2 y} {\d x^2} = \map f y$
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation: differential equations of the second order: $(2)$ Equations of the form $\dfrac {\d^2 y} {\d x^2} = \map f y$