Find a particular solution to the differential equation using the Method of Undetermined Coefficients.

9y'' + 5y' - y = 14

y = c1e^[ (-5 + √61 (/18] + c2e^[ (-5 - √61) / 18] - 14

Step-by-step explanation:

To solve the differential equation

9y'' + 5y' - y = 14 (equation 1)

is to obtain the general equation y = y_c + y_p.

To do that, we need the complimentary function, y_c, and the particular integral, y_p.

To obtain the particular integral, using the method of undetermined coefficients, we need a trial function, y_p, such that

9y_p'' + 5y_p' - y_p = 14 (equation 2)

This works by guessing. When a guess gives a trivial y_p = 0, we then make another guess.

14 is a constant, we guess a constant trial function.

Let y_p = A

y_p' = 0

y_p'' = 0

Substitute these values into equation 2

9 (0) + 5 (0) - (0*x + B) = 14

- B = 14 or B = - 14

y_p = Ax + B = - 14 (equation 3)

To obtain the compliment function, we need to solve the homogeneous part of equation 1.

The homogeneous part is

9y'' + 5y' - y = 0

The auxiliary equation is

9m² + 5m - 1 = 0

Solving the quadratic equation, using the quadratic formula

x = [-b ± √ (b² - 4ac) ]/2a

With a = 9, b = 5, and c = 0

x = [-5 ± √ (5² - 4*9 * (-1)) ]/2 (9)

= [-5 ± √ (25 + 36) ]/18

x1 = (-5 + √61) / 18

x2 = (-5 - √61) / 18

y_c = c1e^ (x1) + c2e^ (x2)

y_c = c1e^[ (-5 + √61 (/18] + c2e^[ (-5 - √61) / 18]

The general solution

y = c1e^[ (-5 + √61 (/18] + c2e^[ (-5 - √61) / 18] - 14