Answer:
0 < t < 5 is the required interval for the differential equation (t - 5)y' + (ln t)y = 6t to have a solution.
Step-by-step explanation:
Given the differential equation
(t - 5)y' + (ln t)y = 6t
and the condition y(1) = 6
We can rewrite the differential equation by dividing it by (t - 5) as
y' + [(ln t)/(t - 5)]y = 6t/(t - 5)
(ln t)/(t - 5) is continuous on the interval (0, 5) and (5, +infinity).
6t/(t - 5) is continuous on (-infinity, 5) and (5, +infinity)
We see that for these expressions, we have continuity at the intervals (0, 5) and (5, +infinity).
But the initial condition is y = 6, when t = 1.
The solution to differential equation is certain to exist at (0, 5)
Which implies that
0 < t < 5
is the required interval.
