Answer:
y = e^(x/3)[0.0016cos(x√7/3) - 1.2056sin(x√7/3)]
Explanation:
The problem is solved by first writing an auxiliary equation
9m² - 6m + 8 = 0
of the differential equation
9y'' - 6y' + 8y = 0.
The auxiliary equation is the solved to obtain the values (1/3 ± i√7/3) of m. These values are then used to obtain the complementary equation
y = e^(x/3)[Acos(x√7/3) + Bsin(x√7/3)]
of the differential equation.
The conditions given are then applied.
1. y(π/2) = -2
Put y = -2 and x = π/2.
This gives an equation in terms of A and B.
2. y'(π/2) = -1
Differentiate y to obtain y', and put y' = -1 and x = π/2.
This gives another equation in terms of A and B.
The two equations obtained here are then solved simultaneously to obtain values for A (0.0016), and B (-1.2056).
These values of A and B are substituted into the complimentary solution obtained earlier to have the desired particular solution.
The step-by-step explanation is shown in the attachment.