Verify that

y $=9$t cos$($t$)9$t

is a solution of the following initial$-$value problem.

t

dydt

$=$ y $+9$t$2$ sin$($t$)$ y$()=0$

y $=9$t cos$($t$)9$t$$

dydt

$=$

$9$cos$($t$)+9$tsin$($t$)9$

LHS$=$t

dydt

$=$ t

$9$cos$($t$)+9$tsin$($t$)9$

$=9$t$2$ sin$($t$)9$t cos$($t$)9$t$=$

$9$tcos$($t$)+9$t$2$sin$($t$)9$t

$+$ y $=$ RHS$,$

so y is a solution of the differential equation. Also

y$()=,$

so the initial condition is satisfied.