Question

x(t)=ft Solve the given initial-value problem: d²/dt² +9x = 5sin(3t), x(0) = 6, x'(0) = -1 You can proceed with solving this initial-value problem.

Answer 1

`\(x(t) = (1)/(18)(2\sin(3t) - 3\cos(3t)) + (31)/(18)e^(-3t) + (19)/(6)\)`the initial conditions
`\(x(0) = 6\) and \(x'(0) = -1\),`we solve for the constants
`\(A\) and \(B\)` in the complementary function. Substituting the values yields
`\(A = (31)/(18)\) and \(B = (19)/(6)\).`

Step-by-step explanation:

The solution to the initial-value problem
`\(x(t) = ft\) with \(d^2/dt^2 + 9x = 5\sin(3t)\), \(x(0) = 6\), and \(x'(0) = -1\)`involves several steps. First, we identify the complementary function by solving the associated homogeneous equation
`\(d^2/dt^2 + 9x = 0\),`resulting in
`\(x_c(t) = A\cos(3t) + B\sin(3t)\).`Then, we determine the particular integral of the non-homogeneous term,
`\(5\sin(3t)\),` which takes the form
`\(x_p(t) = (1)/(18)(2\sin(3t) - 3\cos(3t))\).`Next, we combine the complementary function and particular integral to find the general solution
`\(x(t) = x_c(t) + x_p(t)\).`

Using the initial conditions
`\(x(0) = 6\) and \(x'(0) = -1\),`we solve for the constants
`\(A\) and \(B\)` in the complementary function. Substituting the values yields
`\(A = (31)/(18)\) and \(B = (19)/(6)\).` Therefore, the complete solution for
`\(x(t)\) is \(x(t) = (1)/(18)(2\sin(3t) - 3\cos(3t)) + (31)/(18)e^(-3t) + (19)/(6)\).`

This solution satisfies the initial-value problem by incorporating the general solution with the determined constants, providing the function
`\(x(t)\)` that satisfies the given differential equation along with the specified initial conditions.

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