# Thread: How to convert this 2nd order ODE to a 1st order standard form

1. ## How to convert this 2nd order ODE to a 1st order standard form

Convert to standard form ODE system y' = f(t, y):
(t^2)y'' + sin(y') + 2y − 1 = 0

the goal is to reduce the equation above to be a first order ode.

2. Originally Posted by oxxiissiixxo
Convert to standard form ODE system y' = f(t, y):
(t^2)y'' + sin(y') + 2y − 1 = 0

the goal is to reduce the equation above to be a first order ode.
Hmm...

I know that if you make the substitutions $u_1=y$ and $u_2=\dot{y}$, you will see that $\dot{u_1}=\dot{y}=u_2$ and $\dot{u_2}=\ddot{y}$

Thus, you get the first order system

$\left\{\begin{array}{lcl}
\dot{u_1}=u_2\\ \dot{u_2}=t^{-2}-2u_1t^{-2}-t^{-2}\sin\left(u_2\right)\end{array}\right.$

Does this make sense?

3. thz