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 $y_{p} = \frac{- x}{6} \int \frac{g d x}{x^{2}} + \frac{1}{10 x} \int g d x + \frac{x^{4}}{15} \int \frac{g d x}{x^{5}}$

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

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

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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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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

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Decision Maths

Statistics

Probability and Statistics

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.