Fnd a general solution to the differential equation $y'' - y' - 2y = 0$. Then, use the two solutions you found to write two other linearly independent solutions to the problem. Write a second general solution using your new linearly independent solutions.
We have the characteristic equation:
$r^2 - r - 2 = 0$
$(r-2)(r+1) = 0$
Thus, we have real, distinct roots $r_1 = 2$ and $r_2 = -1$. Our independent solutions are then $c_1e^{r_1x}$ and $c_2e^{r_2x}$ and so our general solution is $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$.
Now, I'm just unsure how I would go about writing two other linearly independent solutions and the second general solution. Can I just substitute two random values for $c_1$ and $c_2$?
Thank you!