Question

"Find particular solutions for each of the following

equations:
(a) y′′ − 2y′ + y = 14x³/2·eˣ
(b) y′′ − 3y′ + 2y = 4/(1 + e⁻ˣ)
(c) x²y′′ − 2xy′ + (x² + 2)y = x³ cos(x)"

Answer 1

To find particular solutions for each of the given differential equations, we need to apply the method of undetermined coefficients.

Step-by-step explanation:

To find particular solutions for each of the given differential equations, we need to apply the method of undetermined coefficients. This method allows us to guess the form of the particular solution based on the form of the nonhomogeneous term.

For equation (a), since the nonhomogeneous term is a product of a polynomial and an exponential function, we can guess a particular solution of the form y_p(x) = (Ax^3 + Bx^2 + Cx + D) e^x. Substitute this solution into the differential equation and solve for A, B, C, and D.

Similarly, for equations (b) and (c), we can make appropriate guesses for the particular solutions and solve for the unknown coefficients.