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This was a question on our test, I know it can be easily solved by Z-transforms but there are no initial conditions specified. In this case, what would be the right approach?
Assume all initial conditions are zero?
Assume $y$ is right-sided, so substitute $k=0$ and $k=-1$ to find $y[2]$, $y[1]$ and $y[0]$?
A system is described by the difference equation $$y[k+2]+1.2y[k+1]+0.32y[k]=(-0.5)^{k+1}$$ Find the value of $y[10]$.
Your intuition that you need initial conditions to fully solve this is correct, so it becomes a problem in test-taking.
Were it me, and were the test proctored by the prof, then I would go to the front of the room and ask. Failing that, I would first solve for $y[10]$ in terms of $y[0]$, $y[1]$ and $y[2]$, then I would point out that as they were not given, then I'm going to assume that they're all equal to one and I'd work out a concrete answer based on that assumption.
This latter approach means that you've (A) given the two most likely solutions that are on the solutions sheet that's to be given to the grader, (B) it establishes that the problem statement wasn't clear, and (C) in the event that the initial conditions were given in some form that you didn't recognize, that's why you're not answering the question as given.
If you were supposed to assume that the initial conditions were zero, you'll get full points. If the prof left off the initial conditions (and isn't a sadistic and/or narcissistic maniac), you'll get full points. If it was a trick question or poorly written and you were given some hint about the initial conditions that you missed (and thus didn't tell us), then it establishes why you didn't answer the question fully, and probably gives you the most partial points possible under the circumstances.
You're right that the question cannot be answered as stated, and there is no "right approach". There are infinitely many solutions of the given difference equation, so without any further assumptions there is no way to compute $y[10]$, or any other value for that matter.
Let $p_1$ and $p_2$ be the two distinct solutions of the characteristic equation
$$p^2+1.2p+0.32=0\tag{1}$$
The homogeneous solution is then given by
$$y_h[k]=Ap_1^k+Bp_2^k\tag{2}$$
with arbitrary constants $A$ and $B$.
The particular solution is given by
$$y_p[k]=Cq^{k+1}\tag{3}$$
with $q=-\frac12$. The constant $C$ is not arbitrary. It must be determined by plugging $(3)$ into the given difference equation:
$$C=\frac{1}{q^2+1.2q+0.32}\tag{4}$$
The complete solution of the difference equation is given by
$$y[k]=y_h[k]+y_p[k]=Ap_1^k+Bp_2^k+Cq^{k+1}\tag{5}$$
with $C$ given by $(4)$. The solution $y[k]$ given by $(5)$ satisfies the given difference equation for any values of $A$ and $B$. In order to determine a specific value of $y[k]$, these constants must be chosen according to some given values $y[k_1]$ and $y[k_2]$ for some values of $k_1$ and $k_2$.
