Second Order ODE/y y'' = (y')^2

Theorem

The second order ODE:

$(1): \quad y y = \paren {y'}^2$

has the general solution:

$y = C_2 e^{C_1 x}$

Proof

Using Solution of Second Order Differential Equation with Missing Independent Variable, $(1)$ can be expressed as:

\(\ds y p \frac {\d p} {\d y}\)

\(=\)

\(\ds p^2\)

where $p = \dfrac {\d y} {\d x}$

\(\ds \leadsto \ \ \)

\(\ds y \frac {\d p} {\d y}\)

\(=\)

\(\ds p\)

\(\ds \leadsto \ \ \)

\(\ds p = \dfrac {\d y} {\d x}\)

\(=\)

\(\ds C_1 y\)

First Order ODE: $x \rd y = k y \rd x$

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds C_2 e^{C_1 x}\)

First Order ODE: $\d y = k y \rd x$

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Problem $1$