The case where t^2*exp(f*t) exists concurs with the case when C(3)<>0 in the particular solution given by eq #17 in my .mw file which I also listed below.
In the case of the Friedlander waveform there is no t^2 term so C(3)=0 which I specified for eq #18 in my .mw file.
I now see your point of the possibility of 3 repeated roots that would yield the t^2 term could occur if one of the roots to the homogeneous characteristic equation is equal to the double REAL root of the nonhomogeneous source term. This not likely to ever happen since the sensor behaves as a harmonic oscillator which means the 2 roots of the characteristic equation are a complex conjugate pair. Nonetheless, I do not think this addresses the discrepancy among the coefficients A(3) to B(1) & A(4) to B(2) respectively for the homogeneous portion of the solution that I point out in eqs #24 through #26.
Your .mw file only addresses the solution method using the LaPlace transform. I have reworked your equations #1 & #5 using the method of undetermined coeffcients that I included in the modified file, Download Laplace.mw. The coefficients, a, b, & c to the characteristic equation are specified to match your equations #1 & #5.
In the case for #1 the eigenfunctions are exp(2t) & exp(-t) which concurs with your solution given by eq #2. Now to account for nonhomogeneous source term I assume the form given by eq #11. After backsubstitution I find that C(3) pertaining to the t^2 term = 0. After resolving both C(1) & C(2) I apply the IC's to resolve B(1) & B(2) which are the coefficients to the homogeneous solution. The results are given by eq #16. Assembly of the total solution is given by eq #17. My observations on the discrepancy are highlighted after eq #17. Similar disrepancies occur for the other case.
So despite matching your coeffcients of equations #1 & #5 the resulting solutions do not concur. Why does the discrepancy exist or where have I gone wrong with my coeffcients?