I need to convert the following stochastic wave equation
$$u_{tt}=u_{xx}+dW,$$
into a system of two equations, the first one is an ODE and the second is an SDE.
Actually, in the following book here, for the solution of the stochastic damped wave equation
which is wrtitten for its solution that it can be rewritten as
So, my try was that $$du=v dt,$$ $$dv=u_{xx}dt+dW.$$
Am I right with that?
Here are you are dealing with an SPDE, so you can't really convert that to a single SDE (any SPDE is actually infinitely many SDEs).
Many references for this equation are listed in "Stochastic Equations in Infinite Dimensions". in 13.21 Wave equations.
See online notes too "The stochastic wave equation" by R.C.Dalang.
For the updated question, indeed by setting $v:=u_{t}$ we get
$$\nu d(u_{t})+u_{t}dt=(u_{xx}+f(u))dt+dW_{t}$$
$$\Rightarrow \nu dv+vdt=(u_{xx}+f(u))dt+dW_{t}$$
$$\Rightarrow dv=\frac{1}{\nu }(-v+u_{xx}+f(u))dt+\frac{1}{\nu }dW_{t}.$$