Second Order ODE/y y'' = (y')^2
Jump to navigation
Jump to search
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 $1$