Final answer:
To solve the differential equation, we propose a form for the particular solution that accounts for the non-homogeneous term and avoid duplicating the solution of the complementary homogeneous equation. After substituting into the equation, we determine the constants by equating coefficients, resulting in the particular solution.
Step-by-step explanation:
To find a particular solution to the differential equation y''+7y'+10y=13te⁵⁴, we need to propose a form for the particular solution, yp, that incorporates the non-homogeneous part 13te⁵⁴. The right-hand side suggests we need a solution that involves t and e⁵⁴. Considering the coefficients of the homogeneous part (7 and 10), it seems likely that we would use a form involving Ate⁵⁴ and Be⁵⁴, where A and B are constants to be determined.
Substitute this assumed form into the differential equation and equate coefficients of like terms to solve for the constants A and B. However, we notice that e⁵⁴ is a solution to the complementary homogeneous equation, so to avoid duplication and because differentiation raises the degree of t, we should propose yp as At²e⁵⁴ + Bte⁵⁴. After calculation, we substitute back to find the particular constants. Once A and B are found, we have found the particular solution yp.