91Ó°ÊÓ

We have a system of differential equations with an exact solution given by \( x(t) = 3 e^{2t} \sin 3t \) and \( y(t) = e^{2t}(\sin 3t - \cos 3t) \). We need to approximate \( x(0.2) \) and \( y(0.2) \) using three different methods (Euler, improved Euler, and Runge-Kutta) and compare these with the exact values.

Use the exact solutions to find \( x(0.2) \) and \( y(0.2) \). Calculate:\[ x(0.2) = 3 \cdot e^{0.4} \cdot \sin(0.6) \approx 1.414 \]\[ y(0.2) = e^{0.4}(\sin(0.6) - \cos(0.6)) \approx 0.229 \]

Use initial values \( x_0 = 0 \), \( y_0 = -1 \). The Euler method update formulas:For the first step:\[ x_1 = x_0 + h(5x_0 - 9y_0) = 0 + 0.1(5 \cdot 0 - 9(-1)) = 0.9 \]\[ y_1 = y_0 + h(2x_0 - y_0) = -1 + 0.1(2 \cdot 0 - (-1)) = -0.9 \]For the second step:\[ x_2 = x_1 + h(5x_1 - 9y_1) = 0.9 + 0.1(5 \cdot 0.9 - 9(-0.9)) = 1.62 \]\[ y_2 = y_1 + h(2x_1 - y_1) = -0.9 + 0.1(2 \cdot 0.9 - (-0.9)) = -0.63 \]

Start with \( x_0 = 0 \), \( y_0 = -1 \).Predictor step:\[ x_{p} = x_0 + h(5x_0 - 9y_0) = 0 + 0.2(5 \cdot 0 - 9(-1)) = 1.8 \]\[ y_{p} = y_0 + h(2x_0 - y_0) = -1 + 0.2(2 \cdot 0 - (-1)) = -0.8 \]Corrector step:\[ x_1 = x_0 + \frac{h}{2}[(5x_0 - 9y_0) + (5x_{p} - 9y_{p})] \]\[ x_1 = 0 + \frac{0.2}{2}[(5 \cdot 0 - 9(-1)) + (5 \cdot 1.8 - 9(-0.8))] = 1.57 \]\[ y_1 = y_0 + \frac{h}{2}[(2x_0 - y_0) + (2x_{p} - y_{p})] \]\[ y_1 = -1 + \frac{0.2}{2}[(2 \cdot 0 - (-1)) + (2 \cdot 1.8 - (-0.8))] = -0.52 \]

Calculate the intermediate steps using the initial values \( x_0 = 0 \), \( y_0 = -1 \).First, calculate intermediate values:\[ k_1 = h(5x_0 - 9y_0) = 0.2(5 \cdot 0 - 9(-1)) = 1.8 \]\[ l_1 = h(2x_0 - y_0) = 0.2(2 \cdot 0 - (-1)) = 0.2 \]\[ k_2 = h\left(5(x_0 + \frac{k_1}{2}) - 9(y_0 + \frac{l_1}{2})\right) \]\[ k_2 = 0.2\left(5(0 + \frac{1.8}{2}) - 9(-1 + \frac{0.2}{2})\right) = 1.95 \]\[ l_2 = h\left(2(x_0 + \frac{k_1}{2}) - (y_0 + \frac{l_1}{2})\right) \]\[ l_2 = 0.2\left(2 \cdot 0.9 - (-1 + 0.1)\right) = 0.31 \]\[ k_3 = h\left(5(x_0 + \frac{k_2}{2}) - 9(y_0 + \frac{l_2}{2})\right) \]\[ k_3 = 0.2\left(5(0 + \frac{1.95}{2}) - 9(-1 + \frac{0.31}{2})\right) = 1.976 \]\[ l_3 = h\left(2(x_0 + \frac{k_2}{2}) - (y_0 + \frac{l_2}{2})\right) \]\[ l_3 = 0.2\left(2 \cdot 0.975 - (-1 + 0.155)\right) = 0.329 \]\[ k_4 = h(5(x_0 + k_3) - 9(y_0 + l_3)) \]\[ k_4 = 0.2(5 \cdot (0 + 1.976) - 9(-1 + 0.329)) = 2.156 \]\[ l_4 = h(2(x_0 + k_3) - (y_0 + l_3)) \]\[ l_4 = 0.2(2 \cdot 1.976 - (-1 + 0.329)) = 0.461 \]Now, calculate the next values:\[ x_1 = x_0 + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) \approx 1.687 \]\[ y_1 = y_0 + \frac{1}{6}(l_1 + 2l_2 + 2l_3 + l_4) \approx -0.494 \]

From Step 2, the exact values are approximately \( x(0.2) = 1.414 \) and \( y(0.2) = 0.229 \). The approximate values from each method are:1. Euler Method: \( x(0.2) = 1.62 \), \( y(0.2) = -0.63 \)2. Improved Euler Method: \( x(0.2) = 1.57 \), \( y(0.2) = -0.52 \)3. Runge-Kutta Method: \( x(0.2) = 1.687 \), \( y(0.2) = -0.494 \)Compare these approximations to see that the Runge-Kutta method gives the results closest to the exact solution.