I need to find solutions for SDE of types: $$ dX_t=a\,dt+b^1\,dW_t^1+b^2\,dW_t^2\\ dX_t=a\,dt+b^1\,dW_t^1+b^2\,X_t\,dW_t^2 $$
As far I've found solutions for SDE with 2-dimensional noise in an amazing book of P. Kloeden - Numerical Solution For Stochastic Differential Equations.
It has a solution for this type of equation: $$ dX_t=aX_t\,dt+b^1X_t\,dW_t^1+b^2X_t\,dW_t^2\\ X_t=X_0\exp\Biggl(\Biggl(a-\frac12((b^1)^2+(b^2)^2)\Biggr)t+b^1W_t^1+b^2W_t^2\Biggr) $$
Is there some generic solution for SDE with 2-dimensional noise like for the General Linear SDE?
Maybe it is possible to use the upper solution here?
UPDATE: As lutzl proposed, the first equation can be reduced to this type: $$dX_t=adt+\sqrt{b_1^2+b_2^2}dW_t$$ Where: $$X_t=at+\sqrt{b_1^2+b_2^2}W_t$$ Is it correct?