# Solving a class of SDEs

Let us try a systematic procedure for solving SDEs which works for a class of SDEs. Let
\begin{align*}
X(t)=a(t)\left[ x_0 + \int_0^t b(s) dB(s) \right] +c(t) \ .
\end{align*}
Assuming $$a$$, $$b$$, and $$c$$ are differentiable, use Ito’s formula to find the equation for $$dX(t)$$ of the form
\begin{align*}
dX(t)=[ F(t) X(t) + H(t)] dt + G(t)dB(t)
\end{align*}
were $$F(t)$$, $$G(t)$$, and $$H(t)$$ are some functions of time depending on $$a,b$$ and maybe their derivatives. Solve the following equations by matching the coefficients. Let $$\alpha$$, $$\gamma$$ and $$\beta$$ be fixed numbers.

1. First consider
$dX_t = (-\alpha X_t + \gamma) dt + \beta dB_t$
with $$X_0 =x_0$$
. Solve this for $$t \geq 0$$
2. Now consider
$dY(t)=\frac{\beta-Y(t)}{1-t} dt + dB(t) ~,~~ 0\leq t < 1 ~,~~Y(0)=\alpha.$
Solve this for $$t\in[0,1]$$.
3. \begin{align*}
dX_t = -2 \frac{X_t}{1-t} dt + \sqrt{2 t(1-t)} dB_t ~,~~X(0)=\alpha
\end{align*}
Solve this for $$t\in[0,1]$$.