How to convert this ODE to a standard form

[oxxiissiixxo]

Homework Statement

Convert to standard form ODE system y0 = f(t, y):
t^2y'' + sin(y') + 2y − 1 = 0

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

Homework Equations

The Attempt at a Solution

I tried to introduced a new variable but the sin(y') seems tricky.

 

[chaoseverlasting]

What is y0? Also, what is y differentiated with respect to.

 

[Unco]

Homework Statement

Convert to standard form ODE system[/color] y0 = f(t, y):
t^2y'' + sin(y') + 2y − 1 = 0

the goal is to reduce the equation above to be a first order ode. Above you said "system".[/color]

The Attempt at a Solution

I tried to introduced a new variable but the sin(y') seems tricky.

I'll assume the derivatives are with respect to t. If we let x(t) = y'(t), then the equation becomes (t^2)x' + sin(x) + 2y - 1 =0, which can be solved for x' in terms of t, x, y. In this way we obtain a system of the form [y', x'] = [f(t,x,y), g(t,x,y)].