Respuesta :
Answer:
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
