You can always rescale the independent variable, for example this way:
independentVariableChange = (a*t + b == x)
dependentVariableChange = (g[t] == f[x])
dependentVariableChange /. Solve[independentVariableChange, x][[1]]
% /. Solve[independentVariableChange, t][[1]]
changeOfVariables =
f -> (Function[x, g[t]] /. Solve[independentVariableChange, t][[1]])
originalDiffEq = f''[x] - f'[x] == x*f[x]
originalDiffEq /. changeOfVariables
% /. Solve[independentVariableChange, x][[1]]