Let's solve the given differential equation using the **Laplace transform** step by step. --- ## **Given:** \[ \frac{d^2y(t)}{dt^2} + 4y(t) = t, \quad y(0) = 1, \quad y'(0) = 2 \] --- ## **Step 1: Take Laplace Transform of Both Sides** Recall: - \(\mathcal{L}\{y(t)\} = Y(s)\) - \(\mathcal{L}\left\{\frac{d^2y}{dt^2}\right\} = s^2Y(s) - sy(0) - y'(0)\) - \(\mathcal{L}\{t\} = \frac{1}{s^2}\) Apply the Laplace Transform: \[ \mathcal{L}\left\{\frac{d^2y}{dt^2}\right\} + 4\mathcal{L}\{y(t)\} = \mathcal{L}\{t\} \] \[ (s^2Y(s) - s y(0) - y'(0)) + 4Y(s) = \frac{1}{s^2} \] --- ## **Step 2: Substitute Initial Conditions** Given \(y(0) = 1\), \(y'(0) = 2\): \[ [s^2Y(s) - s(1) - 2] + 4Y(s) = \frac{1}{s^2} \] \[ s^2Y(s) + 4Y(s) - s - 2 = \frac{1}{s^2} \] --- ## **Step 3: Solve for \(Y(s)\)** Bring \(Y(s)\) terms together: \[ (s^2 + 4) Y(s) = \frac{1}{s^2} + s + 2 \] \[ Y(s) = \frac{\frac{1}{s^2} + s + 2}{s^2 + 4} \] --- ## **Step 4: Simplify \(Y(s)\) (Optional)** Combine into a single fraction: \[ Y(s) = \frac{1 + s^2(s + 2)}{s^2(s^2 + 4)} \] Or, write explicitly: \[ Y(s) = \frac{1}{s^2(s^2 + 4)} + \frac{s}{s^2 + 4} + \frac{2}{s^2 + 4} \] --- ## **Final Answer** \[ \boxed{ Y(s) = \frac{1}{s^2(s^2 + 4)} + \frac{s}{s^2 + 4} + \frac{2}{s^2 + 4} } \] or \[ \boxed{ Y(s) = \frac{1 + s^2(s + 2)}{s^2(s^2 + 4)} } \] This is the Laplace transform of the given differential equation with the initial conditions provided.