DSolve fails on initial value problem of 1st order ODE

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

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

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.

• Jan 22, 2021 at 4:41
• The method in the link does not happen to work in this case, which suggests that this ODE cannot be solve by DSolve. You could use NDSolve, if a numerical solution is satisfactory. Jan 22, 2021 at 4:43

We can solve this equation by series approximation, using AsymptoticDSolveValue if you don't need a numerical solution.

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}]