Let's break down the steps to solve the initial value problem given: ## Step 1: **Express \( h(t) \) Using Heaviside Functions** Given: \[ h(t) = \begin{cases} 34, & \frac{\pi}{5} \leq t < \frac{2\pi}{5} \\ 0, & 0 \leq t < \frac{\pi}{5} \text{ or } t \geq \frac{2\pi}{5} \end{cases} \] We can express this using Heaviside (unit step) functions: \[ h(t) = 34 \left[ u_{c}(t) - u_{d}(t) \right] \] where \( c = \frac{\pi}{5} \) and \( d = \frac{2\pi}{5} \). **Explanation:** The Heaviside function \( u_{c}(t) \) "turns on" at \( t = c \), and \( u_{d}(t) \) "turns off" at \( t = d \), so the expression is nonzero only between \( c \) and \( d \). --- ## Step 2: **Find the Solution Using Laplace Transforms** The differential equation is: \[ y'' + 6y' + 34y = h(t), \quad y(0) = 0, \quad y'(0) = 5 \] Take the Laplace transform of both sides: \[ \mathcal{L}\{y'' + 6y' + 34y\} = \mathcal{L}\{h(t)\} \] This gives: \[ (s^2 Y(s) - s y(0) - y'(0)) + 6(s Y(s) - y(0)) + 34 Y(s) = 34 \left( \frac{e^{-cs}}{s} - \frac{e^{-ds}}{s} \right) \] Plug in the initial conditions (\(y(0) = 0, y'(0) = 5\)): \[ (s^2 Y(s) - 5) + 6s Y(s) + 34 Y(s) = 34 \left( \frac{e^{-cs}}{s} - \frac{e^{-ds}}{s} \right) \] \[ (s^2 + 6s + 34) Y(s) = 5 + 34 \left( \frac{e^{-cs}}{s} - \frac{e^{-ds}}{s} \right) \] So, \[ Y(s) = \frac{5}{s^2 + 6s + 34} + \frac{34}{s^2 + 6s + 34}\left( \frac{e^{-cs}}{s} - \frac{e^{-ds}}{s} \right) \] Now, use inverse Laplace and convolution theorem to get \( y(t) \). **Explanation:** We used Laplace transforms to convert the differential equation into an algebraic equation, making it easier to solve for \( Y(s) \). --- ## **Final Answer:** \[ \boxed{ y(t) = y_0(t) + 34 \int_{c}^{d} G(t-\tau) \, d\tau } \] where: - \( y_0(t) \) is the solution to the homogeneous equation with initial conditions, - \( G(t) \) is the impulse response: \( G(t) = \mathcal{L}^{-1}\left\{\frac{1}{s^2 + 6s + 34}\right\} \), - \( c = \frac{\pi}{5} \), \( d = \frac{2\pi}{5} \). To explicitly write the solution: \[ y(t) = \mathcal{L}^{-1} \left\{ \frac{5}{s^2 + 6s + 34} \right\} + 34 \left[ \int_{c}^{t} G(t-\tau)\, d\tau - \int_{d}^{t} G(t-\tau)\, d\tau \right] \] or using Heaviside functions: \[ y(t) = y_0(t) + 34 \int_0^t G(t-\tau) \left[ u_c(\tau) - u_d(\tau) \right] d\tau \] **Where:** - \( y_0(t) = e^{-3t}\left(\cos(\sqrt{25}t) + \frac{6}{\sqrt{25}}\sin(\sqrt{25}t)\right) \) (after solving the homogeneous part) - \( G(t) = \frac{1}{\sqrt{25}} e^{-3t} \sin(5t) \) --- **Summary of Steps:** 1. Write \( h(t) \) using Heaviside functions. 2. Solve using Laplace transforms and apply convolution for the forcing term. Let me know if you want the explicit expressions for \( y_0(t) \) and the convolution integral!