In essence, linear differential equations with constant coefficients must be solved by degenerate method, and the key is to choose appropriate transformation.

For example, your equation, y''-y=0,

If y'-y=p, then p' = y'-y', and the first-order linear equation p'+p=0 is obtained.

If y'+y=p, then p' = y'+y', we can also get the first-order linear equation p'-p=0.

This is equivalent to the method of deriving eigenvalues. If c is the root of the eigenvalue equation, then p=y'-cy should be chosen.

If you do it your way, then p=y', p'=y'', and then you get the system of equations.

y'=p

p'=y

This order reduction method can be transformed into a first-order linear equation set, and the solution of the equation set is actually the eigenvalue method. By similar transformation, the coefficient matrix is simplified to the standard form and then solved, which is essentially the same as the above method.