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You've already found the solutions of the characteristic equation as $\lambda_1=\frac12$ and $\lambda_2=-\frac14$, so you know that the solution must have the form $$y[n]=c_1\left(\frac12\right)^n+c_2\left(-\frac14\right)^n\tag{1}$$ For the anti-causal solution we know that $y[n]=0$ for $n>0$. The constants $c_1$ and $c_2$ are simply determined by making ...