System of Stochastic Diff Eq How can I solve the system of stochastic differential equation
$$dX_{1}=X_{2}dt+adW_{1}$$ 
$$dX_{2}=-X_{1}dt+bdW_{1}$$
 A: Let $$
X_1(t) = A(t) \sin (t) + B(t) \cos (t)\\
X_2(t) = A(t) \cos (t) - B(t) \sin (t)
$$
then thanks to the Ito fornula:
\begin{align}
dX_1(t) &= dA(t) \sin (t) + A(t) \cos (t) dt 
+ dB(t) \cos (t) - B(t)\sin(t) dt \\
&= X_2(t)dt + dA(t) \sin (t) + dB(t) \cos (t) 
\\
dX_2(t) &= dA(t) \cos (t) - A(t)\sin (t) dt 
- dB(t) \sin (t) - B(t) \cos (t) dt \\
&= -X_1(t) + dA(t) \cos (t) - dB(t) \sin (t) \\
\end{align}
then you are left with
\begin{align}
adW_1(t) &=  dA(t) \sin (t) + dB(t) \cos (t) \\
bdW_1(t) &=  dA(t) \cos (t) - dB(t) \sin (t) \\
\implies 
dA(t) &= \left[ a \sin(t) + b \cos(t) \right] dW_1(t)
 \\
dB(t) &= \left[ a \cos(t) - b \sin(t) \right] dW_1(t) 
\\
A(t) &= A_0 + \int_0^t \left[ a \sin(s) + b \cos(s) \right] dW_1(s)
\\
B(t) &= B_0 + \int_0^t \left[ a \cos(s) - b \sin(s) \right] dW_1(s) 
\end{align}
A: I did that like that
Let $f(t)$ and $g(t)$ be integrating factors
By Ito's Formula
$$d(f(t)X1(t))=f(t)dX1(t)+f'(t)X1(t)dt$$
$$d(f(t)X1(t))=(f(t)X2(t)+f'(t)X1(t))dt+af(t)dW_{1}(t)$$
Similarly
$$d(g(t)X2(t))=(-g(t)X1(t)+g'(t)X2(t))dt+bg(t)dW_{2}(t)$$
Need to find $f(t)$and $g(t)$ such that
$f(t)X2(t)+f'(t)X1(t)=0$ and  $-g(t)X1(t)+g'(t)X2(t)$
So,
$f(t)=sint$ and $g(t)=cost$
therefore
$$d(sintX1(t))=asintdW_{1}(t)$$
$$X1(t)=acosct\int_{0}^{t} sinsdW_{1}(s)$$
and
$$d(CosX2(t))=bcostdW_{2}(t)$$
$$X2(t)=X2(0)sect+bsect\int_{0}^{t} cossdW_{2}(s)$$
Is there anything which I did wrong?
