This finds the general solution. It has only one arbitrary constant because the second equation in the system specifies the relationship between x[t] and y[t].
The general solution is composed of the general solution to the corresponding homogeneous system and a particular solution to the inhomogeneous equation.
Note that there are no degrees of freedom in the solution (that is, there are no arbitrary constants) because z[t] is given algebraically, and thus x[t] and y[t] can be determined uniquely from z[t] using differentiation.
In this example, the algebraic constraint is present only implicitly: all three equations contain derivatives of the unknown functions.
The Jacobian with respect to the derivatives of the unknown functions is singular, so that it is not possible to solve for them.
The differential-algebraic character of this problem is clear from the smaller number of arbitrary constants (two rather than three) in the general solution.
Finally, here is a system with a third-order ODE. Since the coefficients are exact quantities, the computation takes some time.
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