Let's solve the differential equation step-by-step: ### **Given:** \[ \frac{dy}{dx} = 2e^{3x} \] with the initial condition: \[ y(1.03) = 11.88 \] We need to find \( y(0.88) \) and \( y(1.50) \). --- ## **Step 1: Integrate the Differential Equation** \[ \frac{dy}{dx} = 2e^{3x} \] Integrate both sides with respect to \( x \): \[ \int dy = \int 2e^{3x} dx \] The right side: \[ \int 2e^{3x} dx = 2 \int e^{3x} dx = 2 \cdot \frac{1}{3}e^{3x} = \frac{2}{3}e^{3x} \] So, \[ y = \frac{2}{3}e^{3x} + C \] --- ## **Step 2: Apply Initial Condition** Given \( y(1.03) = 11.88 \): \[ 11.88 = \frac{2}{3}e^{3 \times 1.03} + C \] Calculate \( e^{3 \times 1.03} \): \[ 3 \times 1.03 = 3.09 \] \[ e^{3.09} \approx 21.976 \] \[ \frac{2}{3} \times 21.976 \approx 14.65 \] So, \[ 11.88 = 14.65 + C \] \[ C = 11.88 - 14.65 = -2.77 \] --- ## **Step 3: Final Solution** \[ \boxed{y(x) = \frac{2}{3}e^{3x} - 2.77} \] --- ## **Step 4: Evaluate at Given Points** ### (a) \( x = 0.88 \): \[ y(0.88) = \frac{2}{3}e^{3 \times 0.88} - 2.77 \] Calculate: \[ 3 \times 0.88 = 2.64 \] \[ e^{2.64} \approx 14.03 \] \[ \frac{2}{3} \times 14.03 \approx 9.35 \] \[ y(0.88) = 9.35 - 2.77 = \boxed{6.58} \] --- ### (b) \( x = 1.50 \): \[ y(1.50) = \frac{2}{3}e^{3 \times 1.50} - 2.77 \] Calculate: \[ 3 \times 1.5 = 4.5 \] \[ e^{4.5} \approx 90.02 \] \[ \frac{2}{3} \times 90.02 \approx 60.01 \] \[ y(1.50) = 60.01 - 2.77 = \boxed{57.24} \] --- ## **Final Answers:** \[ \boxed{y(0.88) = 6.58} \] \[ \boxed{y(1.50) = 57.24} \] --- **Summary Table:** | x-coordinate | Solution Value | |--------------|---------------| | 0.88 | 6.58 | | 1.50 | 57.24 |