Question

Use the Laplace transform to solve the given initial value problem. $$ y^{\prime \prime}+\omega^{2} y=\cos 2 t, \quad \omega^{2} \neq 4 ; \quad y(0)=1, \quad y^{\prime}(0)=0 $$

Short answer

Question: Determine the solution to the initial value problem $$y^{\prime \prime}+\omega^{2} y=\cos 2 t, \quad \omega^{2} \neq 4 ; \quad y(0)=1, \quad y^{\prime}(0)=0. $$ Answer: The solution to the given initial value problem is $$y(t) = \frac{\omega^2}{2(\omega^2 - 4)}\sin(\omega t) + \frac{1}{2}\sin(2t).$$

Step-by-step solution

Apply the Laplace transform to the given differential equation

To begin solving the initial value problem, we need to apply the Laplace transform to both sides of the given differential equation. The Laplace transform of the given equation is: $$ \mathcal{L}\{y^{\prime\prime}\} + \omega^2\mathcal{L}\{y\} = \mathcal{L}\{\cos 2t\} $$ Using Laplace transform properties: $$ s^{2}Y(s) - sy(0) - y'(0) + \omega^{2}Y(s) = \frac{s}{s^2 + 4} $$ Now substitute the initial conditions \(y(0) = 1\) and \(y'(0) = 0\), and solve for Y(s).

Substitute the initial conditions and solve for Y(s)

Plug in the initial conditions \(y(0) = 1\) and \(y'(0) = 0\) into the equation: $$ s^{2}Y(s) - s + \omega^{2}Y(s) = \frac{s}{s^2 + 4} $$ Next, combine the terms with Y(s): $$ Y(s)(s^{2} + \omega^{2}) = s + \frac{s}{s^2 + 4} $$ Now, solve for Y(s): $$ Y(s) = \frac{s + \frac{s}{s^2 + 4}}{s^{2} + \omega^{2}} $$ $$ Y(s) = \frac{s(s^2 + 4) + s}{(s^2 + \omega^2)(s^2 + 4)} $$

Apply inverse Laplace transform to solve for y(t)

Finally, we apply the inverse Laplace transform to Y(s) to obtain the solution y(t): $$ y(t) = \mathcal{L}^{-1}\left\{ \frac{s(s^2 + 4) + s}{(s^2 + \omega^2)(s^2 + 4)} \right\} $$ Apply partial fraction decomposition to Y(s) and then apply inverse Laplace transform to each term: $$ y(t)=\frac{1}{\omega^2-4}[(\omega^2-4)\mathcal{L}^{-1}\left\{\frac{s}{s^{2}+4} \right\}-\omega^{2} \mathcal{L}^{-1}\left\{\frac{s}{s^{2}+\omega^{2}} \right\}] $$ Finally, evaluate the inverse Laplace transforms of the terms: $$ y(t) = \frac{1}{\omega^2-4}[(\omega^2-4)\frac{1}{2}\sin(2t) - \omega^{2}\frac{1}{\omega}\sin(\omega t)] $$ Simplify the expression: $$ y(t) = \frac{\omega^2}{2(\omega^2 - 4)}\sin(\omega t) + \frac{1}{2}\sin(2t) $$ So the solution to the given initial value problem is: $$ y(t) = \frac{\omega^2}{2(\omega^2 - 4)}\sin(\omega t) + \frac{1}{2}\sin(2t) $$