Question

Determine a suitable form for y(t) if the method of undetermined coefficients is to be used y'''' 2y''' 2y'

Answer 1

A suitable form for y(t) is y(t) = At² + Bt + C +
`e^t`(Dcos(t) + Esin(t)), where A, B, C, D, and E are constants.

Step-by-step explanation:

To determine a suitable form for y(t) when using the method of undetermined coefficients for the given fourth-order linear homogeneous differential equation y'''' - 2y''' + 2y' = 0, we consider the characteristic equation. The characteristic equation associated with the homogeneous part of the differential equation is r⁴ - 2r³ + 2r = 0. Solving this characteristic equation, we find roots with multiplicity, leading to the following complementary function: y_c(t) = (A + Bt + Ct²)
`e^t` + (Dcos(t) + Esin(t))
`e^t`.

To find the particular solution, we look for a particular form that does not overlap with the complementary function. Since the non-homogeneous part involves exponential and trigonometric functions, we choose a particular solution of the form y_p(t) =
`e^t`(Ft² + Gt + H) + Icos(t) + Jsin(t). Substituting this into the original differential equation, we solve for the coefficients F, G, H, I, and J.

After obtaining the values for these coefficients, the final particular solution is added to the complementary function to obtain the overall solution: y(t) = y_c(t) + y_p(t). Therefore, the suitable form for y(t) is y(t) = At² + Bt + C +
`e^t`(Dcos(t) + Esin(t)), where A, B, C, D, and E are constants.