Question

Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. (enter your answer using interval notation.) t(t − 9)y' + y = 0, y(1) = 1

Answer 1

If either $t=0$ or $t=9$, the ODE has the trivial solution $y=0$, but $y(1)=1\neq0$. So any solution to the ODE that satisfies the initial condition can only exist on the interval $0$.