Correct Definition of function in first order equivalent differential equation

Question

When transforming $y''(t)+\sin(y'(t))=\cos(y(t))$ to its first order equivalent: $Y'(t)= F(t, Y)$, I need to provide a rigorous definition to the function $F: \Bbb R\times \Bbb R\times \Bbb R \rightarrow \Bbb R\times \Bbb R $.

I don't know if the second is incorrect:

1. $F(Y)=\pmatrix{Y_2\\\cos(Y_1)-\sin(Y_2)}$ with $Y=\pmatrix{Y_1\\Y_2}$,
2. $F(t,Y)=\pmatrix{y'(t)\\\cos(y(t))-\sin(y'(t))}$ with $Y=\pmatrix{y(t)\\y'(t)}$,
3. $F(t,X)=\pmatrix{X_2\\\cos(X_1)-\sin(X_2)}$ with $Y=\pmatrix{X_1\\X_2}$,

[original Correct Definitions]1

What I did: the first one is ruled out because it takes only one argument and the second is ruled out because $Y(t)$ does not relate to $Y$ but I am not sure.

Thank you for your help.

Answer 1

All the variants are correct in some sense.

• In the full formal sense, only the third is correct, as it has the correct function format and treats all scalar variables as scalar variables. That the variables are named $X$ and not $Y$ is an non-important internal detail.
• The second variant would be correct if the expressions $y(t)$ and $y'(t)$ were treated as simple names for scalar variables, which requires some mental contortion.
• The first is correct for the autonomous ODE system $Y'=F(Y)$, which is a valid interpretation of this ODE, but not in the format required for the task, as you observed.