Final answer:
To find a particular solution to the differential equation 4y'' - y = 2t - t² + 3e³ᵗ, match the form of the nonhomogeneous terms with a trial solution of At + Bt² + Ce³ᵗ and solve for the coefficients by substituting into the equation and matching terms.
Step-by-step explanation:
The differential equation in question is 4y'' - y = 2t - t² + 3e³ᵗ. To find a particular solution, we would typically look for solutions that match the form of the nonhomogeneous part, which is the right side of the equation. In this case, the right side consists of a polynomial part 2t - t² and an exponential part 3e³ᵗ.
Firstly, let's consider the polynomial part. A general form of the polynomial that could work as a particular solution for this part might be At + Bt².
Secondly, the exponential part can be matched with Ce³ᵗ where C is a constant to be determined.
Combining these gives us a trial particular solution of the form At + Bt² + Ce³ᵗ. To find the coefficients A, B, and C, we substitute this trial solution into the left side of the differential equation and match the terms with the right side.
Through a process of differentiation and substitution, we can solve for A, B, and C to get the particular solution that satisfies the given differential equation. Note that this process typically involves finding the first and second derivatives of the trial solution, inserting them into the left-hand side of the equation, and equating coefficients of corresponding powers of t as well as the coefficients of the exponential functions on both sides of the equation.