It looks like the other integral transforms apply linearity as well:
In[1]:= LaplaceTransform[y'[x] + y''[x], x, s]
Out[1]= s LaplaceTransform[y[x], x, s] +
s^2 LaplaceTransform[y[x], x, s] - y[0] - s y[0] - Derivative[1][y][0]
In[2]:= FourierSinTransform[y'[t] + y''[t], t, \[Omega]]
Out[2]= -\[Omega] (FourierCosTransform[y[t],
t, \[Omega]] + \[Omega] FourierSinTransform[y[t], t, \[Omega]] -
Sqrt[2/\[Pi]] y[0])
In[3]:= FourierTransform[y'[t] + y''[t], t, \[Omega]]
Out[3]= FourierTransform[
Derivative[1][y][t] + (y^\[Prime]\[Prime])[t], t, \[Omega]]
In[4]:= FourierCosTransform[y'[t] + y''[t], t, \[Omega]]
Out[4]= -\[Omega]^2 FourierCosTransform[y[t],
t, \[Omega]] + \[Omega] FourierSinTransform[y[t], t, \[Omega]] -
Sqrt[2/\[Pi]] (y[0] + Derivative[1][y][0])
I didn't test other generalities:
http://functions.wolfram.com/GeneralIdentities/11/