Let's work through the differential equation step by step. ### Given Differential Equation \[ y' + \cos(x)y = \cos(x) \] This is a first-order linear differential equation of the form: \[ y' + P(x)y = Q(x) \] where \(P(x) = \cos(x)\) and \(Q(x) = \cos(x)\). --- ## **Step 1: Find the Integrating Factor** The integrating factor, \( \mu(x) \), is: \[ \mu(x) = e^{\int P(x)\,dx} = e^{\int \cos(x)\,dx} = e^{\sin(x)} \] --- ## **Step 2: Multiply Both Sides by the Integrating Factor** \[ e^{\sin(x)}y' + e^{\sin(x)}\cos(x)y = e^{\sin(x)}\cos(x) \] The left side is the derivative of \( e^{\sin(x)}y \): \[ \frac{d}{dx}\left[e^{\sin(x)}y\right] = e^{\sin(x)}\cos(x) \] --- ## **Step 3: Integrate Both Sides** \[ \int \frac{d}{dx}\left[e^{\sin(x)}y\right] dx = \int e^{\sin(x)}\cos(x) dx \] Left side integrates to \( e^{\sin(x)}y \). For the right side, let \(u = \sin(x)\), then \(du = \cos(x)dx\): \[ \int e^{\sin(x)}\cos(x) dx = \int e^u du = e^u = e^{\sin(x)} \] So, \[ e^{\sin(x)}y = e^{\sin(x)} + C \] --- ## **Step 4: Solve for \( y \)** \[ y = 1 + C e^{-\sin(x)} \] --- ## **Step 5: Use Initial Condition** Given \( y(2.49) = 3.86 \): \[ y(2.49) = 1 + C e^{-\sin(2.49)} = 3.86 \] \[ C e^{-\sin(2.49)} = 2.86 \] \[ C = \frac{2.86}{e^{-\sin(2.49)}} = 2.86 e^{\sin(2.49)} \] Compute \( \sin(2.49) \): \[ \sin(2.49) \approx 0.606 \] \[ e^{\sin(2.49)} \approx e^{0.606} \approx 1.833 \] \[ C \approx 2.86 \times 1.833 \approx 5.24 \] --- ## **Step 6: Write Final Solution** \[ y(x) = 1 + 5.24 e^{-\sin(x)} \] --- ## **Step 7: Find \( y(2.56) \) and \( y(2.80) \)** ### For \( x_2 = 2.56 \): \[ \sin(2.56) \approx 0.551 \] \[ e^{-\sin(2.56)} \approx e^{-0.551} \approx 0.576 \] \[ y(2.56) = 1 + 5.24 \times 0.576 \approx 1 + 3.018 \approx \boxed{4.02} \] --- ### For \( x_3 = 2.80 \): \[ \sin(2.80) \approx 0.334 \] \[ e^{-\sin(2.80)} \approx e^{-0.334} \approx 0.716 \] \[ y(2.80) = 1 + 5.24 \times 0.716 \approx 1 + 3.75 \approx \boxed{4.75} \] --- ## **Final Answers** | \( x \) | \( y(x) \) | |-----------|------------| | 2.49 | 3.86 | | 2.56 | 4.02 | | 2.80 | 4.75 | --- ### **Summary** 1. General solution: \( y(x) = 1 + C e^{-\sin(x)} \) 2. Using \( y(2.49) = 3.86 \), found \( C \approx 5.24 \) 3. Answered: - \( y(2.56) \approx 4.02 \) - \( y(2.80) \approx 4.75 \) Let me know if you need more decimal places or further steps!