Consider the second-order initial value problem y''(x)+4y'(x)+5y(x)=0 with y(0)=3,y'(0)=-5. (a) Reformulate the given second-order equation as an equivalent system of two first-order differential equations. (b) Solve the reformulated system over the interval [0,0.2] using the second-order Runge. Kutta'method (RK2) with a step size h=0.1

(a) To reformulate the given second-order equation as an equivalent system of two first-order differential equations, we can introduce a new variable $z(x)$ such that $z(x) = y'(x)$. Then, we can rewrite the equation as:<br /><br />$y'(x) = z(x)$<br />$y''(x) = z'(x)$<br /><br />Substituting these into the original equation, we get:<br /><br />$z'(x) + 4z(x) + 5y(x) = 0$<br /><br />Now, we can express $y(x)$ in terms of $z(x)$ and $y(x)$ as:<br /><br />$y(x) = \frac{z(x)}{5}$<br /><br />So, the equivalent system of two first-order differential equations is:<br /><br />$y'(x) = z(x)$<br />$z'(x) + 4z(x) + 5y(x) = 0$<br /><br />(b) To solve the reformulated system over the interval $[0,0.2]$ using the second-order Runge-Kutta method (RK2) with a step size $h=0.1$, we can follow these steps:<br /><br />1. Initialize the values: $x_0 = 0$, $y_0 = 3$, $z_0 = -5$, and $h = 0.1$.<br />2. For $i = 1, 2,..., 20$ (since $0.2/0.1 = 2$), do the following:<br /> a. Calculate the intermediate values:<br /> $k_1y = y_i + h \cdot z_i$<br /> $k_1z = z_i + h \cdot (4z_i + 5 \cdot k_1y)$<br /> b. Update the values:<br /> $y_{i+1} = y_i + \frac{h}{6} \cdot (k_1y + 2 \cdot k_2y)$<br /> $z_{i+1} = z_i + \frac{h}{6} \cdot (k_1z + 2 \cdot k_2z)$<br /> c. Calculate the error:<br /> $e_y = y_{i+1} - y_i$<br /> $e_z = z_{i+1} - z_i$<br /> d. Update the values:<br /> $y_{i+1} = y_{i+1} + \frac{e_y}{1 + e_y + e_z}$<br /> $z_{i+1} = z_{i+1} + \frac{e_z}{1 + e_y + e_z}$<br /> e. Update $i$ and repeat the process until $i = 20$.<br /><br />After following these steps, we can obtain the approximate solutions for $y(x)$ and $z(x)$ over the interval $[0,0.2]$ using the second-order Runge-Kutta method (RK2) with a step size $h=0.1$.