Final answer:
To determine the proper form of a particular solution for the given differential equation, we need to find the homogeneous solution and a particular solution. The homogeneous solution is found by setting the right-hand side of the equation to zero and solving for y. The particular solution is found by assuming a solution of a specific form and solving for the unknowns.
Step-by-step explanation:
The given differential equation is y'''-6y''+9y' = 5t²et³ᵗ+2cos(3t)+3t.
To determine the proper form of a particular solution, we need to find the homogeneous solution and a particular solution.
The homogeneous solution is found by setting the right-hand side of the equation to zero and solving for y. In this case, we get the characteristic equation r³-6r²+9r = 0. The roots of this equation are r = 3 (triple root).
The particular solution is found by assuming a solution of the form yp = At²et³ᵗ+Bcos(3t)+Ct, where A, B, and C are constants. Plugging this into the original equation and comparing coefficients, we can solve for the unknowns and determine the proper form of the particular solution.