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

**oxxiissiixxo** · January 9th 2009, 05:12 PM

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.

**Chris L T521** · January 9th 2009, 05:23 PM

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

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

Does this make sense?

**oxxiissiixxo** · January 9th 2009, 07:31 PM

thz

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