Excuse the generality of my question. I'm new to the topic of stochastic processes and want to know is their a general form of SDEs that have closed-form solutions?
I saw that $dX_t = dt + 2\sqrt{X_t}dW_t$ has a closed form solution, which can be obtained through Ito's Formula.
A linear SDE is of the form $$(\star) \qquad \mathrm{d}X_t = (a(t)X_t+b(t))\mathrm{d}t + (g(t)X_t + h(t))\mathrm{d}B_t,$$ where $a$, $b$, $g$, $h \colon \mathbb{R}_+ \rightarrow \mathbb{R}$. Its closed form solution might be obtain via integrating factor method as it has been indicated in my comment.
There is a class of reducible SDEs which may be transformed to the form $(\star)$. The results on transforming the SDEs of the form $$(\dagger) \qquad \mathrm{d}X_t = \mu(t, X_t)\mathrm{d}t + \sigma(t, X_t)\mathrm{d}B_t,$$ where $\mu$, $\sigma \colon \mathbb{R}_+ \times \mathbb{R} \rightarrow \mathbb{R}$, into $(\star)$ can be found in Chapter 18, section 18.5 of the book R. L. Schilling and L. Partzsch, "Brownian Motion. An Introduction to Stochastic Processes", De Gruyter.
They use Ito's lemma for a transformed $X_t$, that is, $Z_t=f(t, X_t)$, where $f$ is invertible in the second coordinate. Given some conditions on the coefficients $\mu$ and $\sigma$ you may transform $(\dagger)$ into $(\star)$-type, namely $$\mathrm{d}Z_t = (a(t)Z_t+b(t))\mathrm{d}t + (g(t)Z_t + h(t))\mathrm{d}B_t.$$ The last SDE can be solved explicitly, and so $X_t = f^{-1}(t, Z_t)$. Now depending on the conditions imposed on the coefficients $\mu$ and $\sigma$ you may obtain the explicit form of $f$.
The second reference is Chapter 4, sections 4.2 and 4.3 in P.E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations", Springer. I preferred Schilling's book.
