Final answer:
The best choice for the particular solution to the nonhomogeneous differential equation is option c) x² + 3, as it provides a polynomial that can produce a term proportional to x when differentiated and satisfy the nonhomogeneous term of the equation.
Step-by-step explanation:
To find a particular solution to the nonhomogeneous differential equation y'' - 4y' + 5y = -10x + 5e, we must determine which of the given functions on the right-hand side fills in the blank and complements the equation. What we're looking for is the correct form that can be substituted into the equation, which when differentiated according to the rules of differential equations, will satisfy the equation for some value of the coefficients. Method of Undetermined Coefficients: This technique involves proposing a form for the particular solution based on the form of the nonhomogeneous term (the right-hand side of the differential equation). Given the options provided, the correct form is a polynomial or an exponential function that can compensate for the linear term and the exponential term on the right-hand side.
Checking the given options:
- a) Sin(x) and d) Cos(2x) are trigonometric functions, which do not match the form we need for the given -10x term.
- b) e²⁹ is an exponential term, but it doesn't match the -10x linear term nor does it give a component that coincides with the 5e on the right.
- c) x² + 3 is a polynomial, and it's the best candidate because it can produce a term proportional to x when differentiated twice and combined with other terms of the differential equation.
Thus, option c) x² + 3 is our best choice for the particular solution.