I have the stochastic ode $dX_{t} = tX_{t} + \sigma \cos t X_{t}dW_{t}$, where $W_{t}$ is the Wiener process. I am trying to solve it and not sure if my work is correct. My attempt:
Rearranging, we have $$\frac{dX_{t}}{X_{t}} = t + \sigma \cos t dW_{t}.$$ Then from Ito's Lemma, $dF(X_{t}) = F'(X_{t})dX_{t} + \frac{1}{2}F''(X_{t})(dX_{t})^{2}$ where $F$ is a "nice" function, and putting $F(X_{t}) = \ln X_{t}$, we observe
$$d\ln X_{t} = \frac{dX_{t}}{X_{t}} - \frac{dX_{t}^{2}}{2X_{t}^{2}} $$
And also, after some calculations, we find $dX_{t}^2 = \sigma^{2}X_{t}^2 \cos^{2}tdt$ since $dtdW_{t} = dt^{2} = 0$ and $dW_{t}^{2} = dt$. Then we have that, after some simplifying,
$$d\ln X_{t} = (t - \frac{\sigma^{2}\cos^{2}t}{2})dt + \sigma \cos t dW_{t}$$
Giving us, after integrating both sides:
$$\ln \frac{X_{t}}{X_{0}} = \int_{0}^{t} (u - \frac{\sigma^{2} \cos^{2} u}{2})\, du + \int_{0}^{t} \sigma \cos u\, dW_{u}$$
Can anyone please tell me if my solution is correct so far. Also, clearly the first integral on the right hand side is easily done. What about the second integral? Can it be done and hence can an explicit solution be written for this ode?