User:Egm6321.f09.team3.Kuljevan.Francisco\Homework 1

$$ \begin{align} \dfrac{d}{dt}\left[f\left(Y^{1}(t),t\right)\right] &= \dfrac{\partial}{\partial s}\left[f\left(Y^{1}(t),t\right)\right] \dfrac{d}{dt}\left[Y^{1}(t)\right] + \dfrac{\partial}{\partial t}\left[f\left(Y^{1}(t),t\right)\right]\\ &=\dfrac{\partial}{\partial s}\left[f\left(Y^{1}(t),t\right)\right] \dfrac{d}{dt}\left[Y^{1}(t)\right] + \dfrac{\partial} {\partial t}\left[f\left(Y^{1}(t),t\right)\right]\\ \end{align} $$

Equation (2):

$$ \begin{align} \dfrac{d^2}{dt^2}\left[f\left(Y^{1}(t),t\right)\right] &= \dfrac{d}{dt}\left[\dfrac{\partial}{\partial s}\left(f\left(Y^{1}(t),t\right)\right) \dfrac{d}{dt}\left(Y^{1}(t)\right) + \dfrac{\partial}{\partial s} \left(f\left(Y^{1}(t),t\right)\right)\right]\\ &=\dfrac{\partial}{\partial s} \left[(\dfrac{\partial}{\partial s} \left(f\left(Y^{1}(t),t\right)\right) \dfrac{d}{dt}\left(Y^{1}(t)\right) +\dfrac{\partial}{\partial t}\left(f\left(Y^{1}(t),t\right)\right)\right] \dfrac{d}{dt}\left(Y^{1}(t)\right)+ \dfrac{\partial}{\partial t} \left[\dfrac{\partial}{\partial s}\left(f\left(Y^{1}(t),t\right)\right) \dfrac{d}{dt}\left(Y^{1}(t)\right) + \dfrac{\partial}{\partial t}\left(f\left(Y^{1}(t),t\right)\right)\right]\\ &=\dfrac{\partial^2}{\partial s^2}\left(f(Y^{1},t)\right) \left[\dfrac{d}{dt}\left(Y^{1}(t)\right)\right]^2 + \dfrac{\partial^2}{\partial s \partial t}\left(f\left(Y^{1},t\right)\right) \dfrac{d}{dt}\left(Y^{1}(t)\right) + \dfrac{\partial^2}{\partial s \partial t}\left(f\left(Y^{1},t\right)\right) \dfrac{d}{dt}\left(Y^{1}(t)\right) + \dfrac{\partial^2}{\partial t^2}\left(f(Y^{1},t)\right) + \dfrac{\partial}{\partial s}\left(f(Y^{1},t)\right) \dfrac{d}{dt}\left[\dfrac{d}{dt}\left(Y^{1}(t)\right)\right]\\ &=\dfrac{\partial^2}{\partial s^2}\left(f(Y^{1},t)\right)\left[\dfrac{d}{dt}\left(Y^{1}(t)\right)\right]^2 +2\cdot \dfrac{\partial^2}{\partial s \partial t}\left(f\left(Y^{1},t\right)\right) \dfrac{d}{dt}\left(Y^{1}(t)\right) + \dfrac{\partial^2}{\partial t^2}\left(f(Y^{1},t)\right) + \dfrac{\partial}{\partial s}\left(f(Y^{1},t)\right) + \dfrac{d^{2}}{dt^{2}}\left(Y^{1}(t)\right) \end{align} $$