The Ornstein-Uhlenbeck process with mean reversion level 0 is defined as follows:

$$dX_t=a X_t dt + \sigma_1 d W_{1t}. \tag{1} $$

Geometric Brownian motion is defined as follows:

$$dX_t= a X_t dt + \sigma_2 X_t d W_{2t}. \tag{2} $$

Hence, the two processes differ only in the second term, which I call noise term.

Is the SDE that contains both noise terms:

$$dX_t=a X_t dt + \sigma_1 d W_{1t} + \sigma_2 X_t d W_{2t}, \tag{3} $$ where $W_{1t}$ and $W_{2t}$ are independent Wiener processes,

valid? If yes, what is the solution?

The only approach I am familiar with to solve SDEs is to use the formula in "Introduction to stochastic integration" by Kuo, Hui-Hsiung on page 233. While I was successful in re-deriving the solutions of both the Ornstein-Uhlenbeck process and Geometric Brownian motion, I was not able to fit the process of interest (the mixture) into the formula.