Q7E Find a general solution to the C... [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: Q7E (page 341)

Find a general solution to the Cauchy-Euler equation $x^{3} y^{'''} - 3 x^{2} y^{''} + 6 x y^{'} - 6 y = x^{- 1}, x > 0,$
given that ${x, x^{2}, x^{3}}$ is a fundamental solution set for the corresponding homogeneous equation

Short Answer

Expert verified

The general solution is $y = C_{1} x + C_{2} x^{2} + C_{3} x^{3} - \frac{1}{24 x}$

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 complementary solution

It is given that $x^{3} y^{'''} - 3 x^{2} y^{''} + 6 x y^{'} - 6 y = \frac{1}{x}$

Also, $\{x, x^{2}, x^{3}\}$ is the fundamental solution

The complementary solution is as follows:

$y_{c} = C_{1} x + C_{2} x^{2} + C_{3} x^{3}$

Now, find out the particular solution of the given differential equation.

Calculate Wornkians

Calculate the Wronskian of the above set as follows:

$W [x, x^{2}, x^{3}] = |\begin{matrix} x & x^{2} & x^{3} \\ 1 & 2 x & 3 x^{2} \\ 0 & 2 & 6 x \end{matrix}| = 12 x^{3} - 6 x^{3} - 6 x^{3} + 2 x^{3} = 2 x^{3}$

The value of wronkians $W_{1}, W_{2}, W_{3}$ is:

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

For particular solution

Evaluate the particular solution

$y^{'''} - \frac{3}{x} y^{''} + \frac{6 y^{'}}{x^{2}} - \frac{6 y}{x^{3}} = \frac{1}{x^{4}} y_{p} = x 鈭� \frac{x^{4} (\frac{1}{x^{4}})}{(\frac{x^{4}}{x^{4}}) 2 x^{3}} d x + x^{2} 鈭� \frac{- 2 x^{3} (\frac{1}{x^{4}})}{(\frac{x^{4}}{x^{4}}) 2 x^{3}} + x^{3} 鈭� \frac{x^{2} (\frac{1}{x^{4}})}{2 x^{3}} d x y_{p} = \frac{- x}{4 x^{2}} + x^{2} 鈭� \frac{- 2}{2 x^{4}} d x + x^{3} 鈭� \frac{d x}{2 x^{5}}$

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91影视

Find a general solution to the Cauchy-Euler equation $x^{3} y^{'''} - 3 x^{2} y^{''} + 6 x y^{'} - 6 y = x^{- 1}, x > 0,$
given that ${x, x^{2}, x^{3}}$ is a fundamental solution set for the corresponding homogeneous equation

Short Answer

Step by step solution

Definition

Find complementary solution

Calculate Wornkians

For particular solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

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Statistics

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91影视

Find a general solution to the Cauchy-Euler equation x3y'''-3x2y''+6xy'-6y=x-1,x>0,given that{x,x2,x3}is a fundamental solution set for the corresponding homogeneous equation

Short Answer

Step by step solution

Definition

Find complementary solution

Calculate Wornkians

For particular solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Logic and Functions

Discrete Mathematics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

Find a general solution to the Cauchy-Euler equation $x^{3} y^{'''} - 3 x^{2} y^{''} + 6 x y^{'} - 6 y = x^{- 1}, x > 0,$
given that ${x, x^{2}, x^{3}}$ is a fundamental solution set for the corresponding homogeneous equation