Final answer
(a) The general solution to the corresponding homogeneous equation is yh(t) = c1e²t + c2e³t.
The particular solution to the nonhomogeneous equation using the method of undetermined coefficients is yp(t) = At² + Bt + Ce³t + Dsin(t) + Ecos(t).
(c) A suitable form for a particular solution of the nonhomogeneous equation g(t) = t² + 2t + te²t + sin³t is yp(t) = At³ + Bt² + Ct + De²t + Ecos(t) + Fsin(t).
Step-by-step explanation:
For the homogeneous equation (a), the general solution involves finding the roots of the characteristic equation derived from the differential equation. In this case, the roots are r = 2 and r = 3, resulting in the general solution yh(t) = c1e²t + c2e³t.
For part (b), solving the nonhomogeneous equation involves finding a particular solution using the method of undetermined coefficients. To solve for the particular solution, you assume a form for yp(t) that includes terms similar to the nonhomogeneous function. Then, differentiate and substitute this form into the original equation, solving for the undetermined coefficients to obtain the particular solution yp(t) = At² + Bt + Ce³t + Dsin(t) + Ecos(t).
Finally, for part (c), given a new g(t), you adjust the form of the particular solution yp(t) accordingly. For g(t) = t² + 2t + te²t + sin³t, the particular solution's form would involve terms for each element in g(t), leading to yp(t) = At³ + Bt² + Ct + De²t + Ecos(t) + Fsin(t), without solving for the constants A, B, C, D, E, or F.