Let `y=y(t)` be a solution to the differential equation `y^(prime)+2t y=t^2,` then `16 ("lim")_(t to oo)y/t` is_______

Answer

8

Explanation

`(dy)/(dx)+2ty=t^(2)`
I.F. `e^(t^(2))`
Thus, solution is `y.e^(t^(2))=int(t^(2)e^(t^(2))dt)=1/2intt.(2t.e^(t^(2)))dt`
`therefore y.e^(t^(t))=t.(e^(t^(2))/2-1/2inte^(t^(2))dt)+C`
`therefore y=t/2-e^(-t^(2))int(e^(t^(2))/2dt +Ce^(-t^(2)))`
`therefore underset(t to infty)"lim"y/t = 1/2 - underset(t to infty)"lim"(inte^(t^(2))/2)/(te^(t^(2))+C/(t.e^(t^(2))))=1/2`