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 can introduce a new variable

such that

. Then, we can rewrite the equation as:

Substituting these into the original equation, we get:

Now, we can express

in terms of

and

as:

So, the equivalent system of two first-order differential equations is:

(b) To solve the reformulated system over the interval

using the second-order Runge-Kutta method (RK2) with a step size

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

,

, and

.2. For

(since

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

b. Update the values:

c. Calculate the error:

d. Update the values:

e. Update

and repeat the process until

.After following these steps, we can obtain the approximate solutions for

and

over the interval

using the second-order Runge-Kutta method (RK2) with a step size

.

Soru