Q10E Given that {x,x-1,x4}聽聽is a fu... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 6: Q10E (page 341)

Given that ${x, x^{- 1}, x^{4}}$ is a fundamental solution set for the homogeneous equation corresponding to the equation $x^{3} y^{'''} - x^{2} y^{''} - 4 x y^{'} + 4 y = g (x), x > 0,$ determine a formula involving integrals for a particular solution.

Short Answer

Expert verified

The particular solution is $y_{p} = \frac{- x}{6} 鈭� \frac{g d x}{x^{2}} + \frac{1}{10 x} 鈭� g d x + \frac{x^{4}}{15} 鈭� \frac{g d x}{x^{5}}$

Step by step solution

Definition

Variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.

Find Wronkians

Consider the differential equation $x^{3} y^{'''} - x^{2} y^{''} - 4 x y^{'} + 4 y = g$

The fundamental set of solution is $\{x, x^{- 1}, x^{4}\}$

$W [\begin{matrix} x & x^{- 1} & x^{4} \end{matrix}] = |\begin{matrix} x & \frac{1}{x} & x^{4} \\ 1 & \frac{- 1}{x^{2}} & 4 x^{3} \\ 0 & \frac{2}{x^{3}} & 12 x^{2} \end{matrix}| = x (- 12 - 8) - (12 x - 2 x) = - 30 x$

Wronkians

The value of Wronkians $W_{1}, W_{2}, W_{3}$ is given by:

$W_{1} = (- 1)^{3 - 1} W [\begin{matrix} x^{- 1} & x^{4} \end{matrix}] = |\begin{matrix} \frac{1}{x} & x^{4} \\ \frac{- 1}{x^{2}} & 4 x^{3} \end{matrix}| = 5 x^{2} W_{2} = (- 1)^{3 - 2} W [\begin{matrix} x & x^{4} \end{matrix}] = - 3 x^{4} W_{3} = (- 1)^{3 - 3} W [\begin{matrix} x & x^{- 1} \end{matrix}] = |\begin{matrix} x & x^{- 1} \\ 1 & \frac{- 1}{x^{2}} \end{matrix}| = \frac{- 2}{x}$

The particular solution is

$y_{p} = x 鈭� \frac{5 x^{2} (\frac{g}{x^{3}}) d x}{- 30 x} + \frac{1}{x} 鈭� \frac{- 3 x^{4} 脳 (\frac{g}{x^{3}}) d x}{- 30 x} + x^{4} 鈭� \frac{\frac{- 2}{x} (\frac{g}{x^{3}}) d x}{- 30 x} y_{p} = \frac{- x}{6} 鈭� \frac{g d x}{x^{2}} + \frac{1}{10 x} 鈭� g d x + \frac{x^{4}}{15} 鈭� \frac{g d x}{x^{5}}$

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91影视

Given that ${x, x^{- 1}, x^{4}}$ is a fundamental solution set for the homogeneous equation corresponding to the equation $x^{3} y^{'''} - x^{2} y^{''} - 4 x y^{'} + 4 y = g (x), x > 0,$ determine a formula involving integrals for a particular solution.

Short Answer

Step by step solution

Definition

Find Wronkians

Wronkians

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91影视

Given that {x,x-1,x4}is a fundamental solution set for the homogeneous equation corresponding to the equationx3y'''-x2y''-4xy'+4y=g(x),x>0,determine a formula involving integrals for a particular solution.

Short Answer

Step by step solution

Definition

Find Wronkians

Wronkians

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Given that ${x, x^{- 1}, x^{4}}$ is a fundamental solution set for the homogeneous equation corresponding to the equation $x^{3} y^{'''} - x^{2} y^{''} - 4 x y^{'} + 4 y = g (x), x > 0,$ determine a formula involving integrals for a particular solution.