Chapter 1: Q5.3-25E (page 1)

Using the Runge–Kutta algorithm for systems with h = 0.05, approximate the solution to the initial value problem $y''' + y'' + y^{2} = t; y (0) = 1, y' (0) = 0, y'' (0) = 1$ at t=1.

Short Answer

Expert verified

The result can get by the Runge-Kutta method and the result is y(1)=1.25958

Step by step solution

Transform the equation

Here the equation is $y''' + y'' + y^{2} = t .$

The system can be written as:

$x_{1} = y (t) x_{2} = y' = x y'' = x_{3}$

The transform equation is:

$x'_{1} = x_{2} x'_{2} = x_{3} x'_{3} = t - x_{3} - {x^{2}}_{1}$

The initial conditions are:

$x_{1} (0) = 1 x_{2} (0) = 1 x_{3} (0) = 1$

Apply the Runge-Kutta method

For h=1

$k_{1, 1} = {hx}_{2} k_{2, 1} = {hx}_{3} k_{3, 1} = h (t - x_{3} - {x^{2}}_{1}) k_{1, 2} = h (x_{2} + \frac{k_{2, 1}}{2}) k_{2, 2} = h (x_{3} + \frac{k_{3, 1}}{2}) k_{3, 2} = h (t + \frac{h}{2} - x_{3} - \frac{k_{3, 1}}{2} - {(x_{1} + \frac{k_{1, 1}}{w})}^{2}) k_{1, 3} = h (x_{3} + \frac{k_{2, 2}}{2}) k_{2, 3} = h (x_{3} + \frac{k_{3, 2}}{2}) k_{3, 3} = h (t + \frac{h}{2} - x_{3} - \frac{k_{3, 2}}{2} - {(x_{1} + \frac{k_{1, 2}}{w})}^{2}) k_{1, 4} = h (x_{2} + k_{2, 3}) k_{2, 4} = h (x_{3} + k_{3, 3}) k_{3, 4} = h (t + h - x_{3} - k_{3, 3} - (x_{1} + k_{1, 3})^{2})$

For,

$t_{0} = 0 a_{1} = 0 a_{2} = 0 a_{3} = 1 x_{1} (1, 1) = 1 . 29167 x_{2} (1, 1) = 0 . 28125$

Repeating the same procedure for $h = 2^{- 1,}, 2^{- 2}, 2^{- 3}$ .

n	h	$x_{1}$	$x_{2}$	$x_{3}$ $x_{3}$
0	1.0	1.29167	0.28125	0.03125
1	0.5	1.26039	0.34509	-0.06642
2	0.25	1.25960	0.346996	-0.06957
3	0.125	1.25958	0.34704	-0.06971

Hence, y (1) =1.25958

This is the required result.

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Using the Runge–Kutta algorithm for systems with h = 0.05, approximate the solution to the initial value problem $y''' + y'' + y^{2} = t; y (0) = 1, y' (0) = 0, y'' (0) = 1$ at t=1.

Short Answer

Step by step solution

Transform the equation

Apply the Runge-Kutta method

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Using the Runge–Kutta algorithm for systems with h = 0.05, approximate the solution to the initial value problem y'''+y''+y2=t;y(0)=1,y'(0)=0,y''(0)=1 at t=1.

Short Answer

Step by step solution

Transform the equation

Apply the Runge-Kutta method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Statistics

Calculus

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Using the Runge–Kutta algorithm for systems with h = 0.05, approximate the solution to the initial value problem $y''' + y'' + y^{2} = t; y (0) = 1, y' (0) = 0, y'' (0) = 1$ at t=1.