Question

(1 pt) let t2y′′−5ty′+5y=0. find all values of r such that y=tr satisfies the differential equation for t>0. if there is more than one correct answer, enter your answers as a comma separated list.

Answer 1

Let
$y = t^r \\ \\ y' = rt^{r-1} \\ \\ y'' = r(r-1) t^{r-2}$

Substitute into differential equation:
$(r^2 -r) t^r - (5r) t^r + (5) t^r = 0$

Factor out t^r and solve for 'r':
$r^2 - r -5r +5 = 0 \\ \\ r^2 - 6r+5 = 0 \\ \\ (r-5)(r-1) = 0 \\ \\ r = 1, r=5$