I have a $2^\text{nd}$ ODE:
$ \begin{cases}{d^2u \over dt^2} =5tu+\sin \left({du\over dt}\right)\\[5 pt] u(0)=1\\[5 pt] {du\over dt}(0)=0 \end{cases} $
I was reading my notes and it asked to write the $2^\text{nd}$ order ODE as a system of $1^\text{st}$ order ODEs. And then to construct a forward euler discretisation of the ODE with step size $\tau =1/2$ and interval $[0,2]$.
What was done in the notes was:
$\begin{align} \text{Let }&v={du \over dt}\\ &{dv \over dt}={d^2u \over dt^2}\\ \implies &{dv \over dt}=5tu+\sin v, \ v(0)=0. \end{align} $
I understood the above, but I'm not sure what was done after that. Could someone explain to me what was done below? Let
$ w= \left( \begin{matrix} u \\ v \end{matrix} \right)\\ \text{then } {dw \over dt}=f(t,w), \;\;\;\;\;\; w(0)=w_0 \\ \text{where } f(t,w)=\left( \begin{matrix} v \\ 5tu+\sin v \end{matrix} \right) \text{ and } w_0=\left( \begin{matrix} 1 \\ 0 \end{matrix} \right)$
Continuing on from there, how does the following work? In particular how does
$ f(t_0, W^0)= \left( \begin{matrix} V^0 \\ 5\cdot 0 \cdot U^0 + \sin V^0 \end{matrix} \right) = \left( \begin{matrix} 0 \\ 0 \end{matrix} \right)$
Forward euler for the $1^\text{st}$ order system: Given $W^0=w_0$, find $W^{n+1}$ such that $W^{n+1}=W^n+\tau f(t_n,w)$
$n=0 \implies $W^0= \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \\f(t_0, W^0)= \left( \begin{array}{c} V^0 \\ 5\cdot 0 \cdot U^0 + \sin V^0 \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right) \\ \implies W^1 = W^0 +\tau \left( \begin{array}{c} 0 \\ 0 \end{array} \right) \implies W^1 = W^0 $