<s>[INST] What does it mean for an ODE to be conservative? What does it mean for an ODE to be conservative?
For example, I already read somewhere that the equation
$$w\cdot y''-y+y^{2k+1}=0,$$
with $w>0$ and $k\in \mathbb{N}$ constants fixeds, is conservative. In practice, what does this mean?
 [/INST] Given the differential equation
$wy'' - y + y^{2k + 1} = 0, \tag 1$
we may multiply it through by $y'$:
$wy'y'' - yy' + y^{2k + 1}y' = 0, \tag 2$
and observe that
$\left ( \left ( \dfrac{w}{2} y' \right )^2 \right )' = wy''y', \tag 3$
and
$\left ( -\dfrac{y^2}{2} + \dfrac{y^{2k + 2}}{2k + 2} \right )' = - yy' + y^{2k + 1}y'; \tag 4$
then (2) may be written
$\left ( \left ( \dfrac{w}{2} y' \right )^2 \right )' + \left ( -\dfrac{y^2}{2} + \dfrac{y^{2k + 2}}{2k + 2} \right )' = 0, \tag 5$
or
$\left ( \left ( \dfrac{w}{2} y' \right )^2 -\dfrac{y^2}{2} + \dfrac{y^{2k + 2}}{2k + 2} \right )' = 0; \tag 6$
thus,
$\left ( \dfrac{w}{2} y' \right )^2 -\dfrac{y^2}{2} + \dfrac{y^{2k + 2}}{2k + 2} = C, \; \text{a constant} \tag 7$
along the solution curves $(x, y(x))$ of (1).  That is, the quantity on the left of (7) is a conserved quantity of the equation(1); hence we deem (1) a conservative ordinary differential equation, since the function
$F(y, y') = \left ( \dfrac{w}{2} y' \right )^2 -\dfrac{y^2}{2} + \dfrac{y^{2k + 2}}{2k + 2} \tag 8$
is invariant in value on the solution curves.
In general, a conservative second-order equation or system is one for which a function such as $F(y, y')$ exists.  For these $F(y, y')$, it follows that
$\dfrac{\partial F(y, y')}{\partial y} y' + \dfrac{\partial F(y, y')}{\partial y'}y'' = \dfrac{dF(y, y')}{dx} = 0, \tag 9$
and thus $y(x)$ satisfies the deifferential equation (9).  We may in fact invoke these principles to construct differential equations corresponding to many different $F(y, y')$; for example, if
$F(y, y') = \cos y + e^{y'}, \tag{10}$
then we have
$- y' \sin y + e^{y'}y'' = 0, \tag{11}$
or
$y'' = e^{-y'}y'\sin y. \tag{12}$
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