Question

Solve the initial value problem: 9y′′−6y′+8y=0, y(π/2)=−2, y′(π/2)=−1. Give your answer as y=... . Use x as the independent variable.

Answer 1

Answer:

y = e^(x/3)[0.0016cos(x√7/3) - 1.2056sin(x√7/3)]

Step-by-step 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.