Question

Use Laplace transforms to solve the following initial value problem. x′′+25x=sin6t,x(0)=0,x′(0)=0 x(t)=

Answer 1

`\(x(t) = (1)/(170)\sin(6t) - (6)/(170) \sin(5t)\)`This solution represents the combination of sinusoidal functions that satisfy the given initial value problem.

Explanation:

To solve the given initial value problem using Laplace transforms, first, we apply the Laplace transform to the differential equation
`\(x'' + 25x = \sin(6t)\)`and the initial conditions
`\(x(0) = 0\) and \(x'(0) = 0\).`

After rearranging and applying the initial conditions, we derive the transformed equation in terms of the Laplace variable
`\(s\),`which gives us the equation in terms of
`\(X(s)\). Solving for \(X(s)\)`yields a partial fraction decomposition.

Through inverse Laplace transform and using trigonometric identities, the solution
`\(x(t)\)` can be expressed as a combination of sine functions with different frequencies, amplitudes, and phase shifts, corresponding to the coefficients obtained from the partial fraction decomposition.

The final solution for
`\(x(t)\) is \(x(t) = (1)/(170)\sin(6t) - (6)/(170) \sin(5t)\).` This solution represents the combination of sinusoidal functions that satisfy the given initial value problem.