Use the Method of Variation of Parameters to Solve the Following

Multiple Choice

Use the method of variation of parameters to solve the following third-order nonhomogeneous differential equation:

A) $y=C_{1} e^{t}+C_{2} t e^{t}+C_{3} t^{2} e^{t}-\frac{2}{5} t^{\frac{5}{2}} e^{t}$
B) $y=C_{1} e^{t}+C_{2} t e^{t}+C_{3} t^{2} e^{t}+\frac{8}{105} t^{\frac{7}{2}} e^{t}$
C) $y=C_{1} e^{t}+C_{2} t e^{t}+C_{3} t^{2} e^{t}+\left(\frac{1}{7} t^{\frac{7}{2}}-\frac{2}{5} t^{\frac{7}{2}}\right) e^{t}$
D) $y=C_{1} e^{-t}+C_{2} t e^{-t}+C_{3} t^{2} e^{-t}+\frac{8}{105} t^{\frac{7}{2}} e^{t}$
E) $y=C_{1} e^{-t}+C_{2} t e^{-t}+C_{3} t^{2} e^{-t}+\left(\frac{1}{7} t^{\frac{7}{2}}-\frac{2}{5} t^{\frac{7}{2}}\right) e^{t}$
F) $y=C_{1} e^{-t}+C_{2} t e^{-t}+C_{3} t^{2} e^{-t}-\frac{2}{5} t^{\frac{5}{2}} e^{t}$