Problem: Approximate the expectation function of the solution of SDE $dX_t = \Bigl[\bigl(\frac1t-\frac1{t^2}+\frac2{t^3}\bigr)+\ln(t)\bigl(\frac1{t^2}-\frac4{t^3}\bigr)-\tfrac12\ln(t)\bigl(1-\frac1t+\frac2{t^2}\bigr)\Bigr]\cos(B_t)\,dt - \ln(t)\bigl(1-\frac1t+\frac2{t^2}\bigr)\sin(B_t)\,dB_t$ in the limit $ t > 0$ Please provide your answer in LaTeX \boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.
Solution: $\boxed{e^{-\frac{t}{2}} \ln(t) (1 - \frac{1}{t} + \frac{2}{t^2})}$