DSolve fails on initial value problem of 1st order ODE

I’ve been trying to solve this initial value problem using DSolve[]:

$$
frac{dy}{dt}=1+tspace sin(tspace y),quad y(0)=0, quad t=[0,2]
$$

ClearAll[y, t]
eq1 := {y'[t] == 1 + t *Sin[t y[t]], y[0] == 0};
DSolve[eq1, y[t], {t, 0, 2}]

All I get is the Inverse function error.

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

The documentation suggests it has to do with the sine function but I’m not sure how to by-pass it.

Any help would be appreciated.

ClearAll[y, t]
eq1 := {y'[t] == 1 + t*Sin[t y[t]], y[0] == 0};
sol = AsymptoticDSolveValue[eq1, y[t], {t, 0, 10}]

(*t + t^4/4 + t^7/28 - t^8/48 + t^10/280*)

Plot[sol, {t, 0, 1}]