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

Question

Answer 1

(a) To reformulate the given second-order equation as an equivalent system of two first-order differential equations, we introduce a new variable

such that

. Then, we can rewrite the given equation as:

Substituting these into the original equation, we get:

Now, we have a system of two first-order differential equations:

(b) To solve the reformulated system over the interval

using the second-order Runge-Kutta method

with a step size

, we can follow these steps:1. Initialize the values:

,

, and

.2. For

, where

, do the following: a. Calculate the intermediate values:

b. Update the values:

c. Calculate the error:

d. Update the values:

e. Update the value of

:

After performing these steps for

, we will have the approximate solutions for

and

over the interval

using the second-order Runge-Kutta method

with a step size

.

Soru