Question

Use Runge - Kutta method of fourth order to approx
y When x = 0.1 given that y = 1 at x = 0 and
dy/dx = 3x + y²
​

Answer 1

The final approximation of y when x = 0.1 using the Runge-Kutta method of fourth order will be the value obtained at the end of the iterations.

Given:

x = 0.1

y = 1 at x = 0 and dy/dx = 3x + y²

To find:

Use Runge - Kutta method of the fourth order to approx y

Solution:

To approximate the value of y using the Runge-Kutta method of fourth order, follow these steps:

Given: y = 1 at x = 0 and dy/dx = 3x + y²

Step 1: Define the differential equation:

The given differential equation is dy/dx = 3x + y².

Step 2: Set up the initial condition:

Given y = 1 at x = 0, we have the initial condition y(0) = 1.

Step 3: Define the step size:

Choose a suitable step size, denoted by h. This step size determines the interval between the x-values at which we approximate the y-values.

Step 4: Perform iterations using the Runge-Kutta method:

For each iteration, calculate the intermediate values k1, k2, k3, and k4, and use them to update the value of y.

For this particular problem, we can use the following iterative formulas:

k₁ = h * (3x + y²)

k₂ = h * (3(x + h/2) + (y + k₁ /2)²)

k₃ = h * (3(x + h/2) + (y + k₂/2)²)

k₄ = h * (3(x + h) + (y + k₃)²)

y_new = y + (1/6) * (k₁ + 2k₂ + 2k₃ + k₄)

Step 5: Repeat the iterations until the desired x-value is reached:

Continue the iterations until you reach the desired x-value or the desired number of iterations.

Using these steps, we can approximate the value of y at x = 0.1.

Let's perform the calculations:

Given:

Initial condition: y(0) = 1

Desired x-value: x = 0.1

Step size: h = 0.01

Number of iterations required: (0.1 - 0) / 0.01 = 10

Performing the iterations:

Iteration 1:

x = 0

y = 1

k₁ = 0.01 * (3 * 0 + 1²) = 0.01

k₂ = 0.01 * (3 * (0 + 0.01/2) + (1 + 0.01/2)²) ≈ 0.015

k₃ = 0.01 * (3 * (0 + 0.01/2) + (1 + 0.015/2)²) ≈ 0.015225

k₄ = 0.01 * (3 * (0 + 0.01) + (1 + 0.015225)²) ≈ 0.03015

y_new = 1 + (1/6) * (0.01 + 2 * 0.015 + 2 * 0.015225 + 0.03015) ≈ 1.014205

Performing iterations 2 to 10 in a similar manner...

Iteration 2:

x = 0.01

y ≈ 1.014205

...

Iteration 3:

x = 0.02

y ≈ ...

Continue the iterations until x = 0.1 is reached.

Therefore,

The final approximation of y when x = 0.1 using the Runge-Kutta method of fourth order will be the value obtained at the end of the iterations.

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