Q4E In Problems 1-6, use the method ... [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: Q4E (page 341)

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} - 3 y^{''} + 3 y^{'} - y = e^{x}$

Short Answer

Expert verified

The particular solution is $y_{p} = \frac{x^{3} e^{x}}{6}$

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

The given equation is: $y^{'''} - 3 y^{''} + 3 y^{'} - y = e^{x}$

The auxiliary equation is $m^{3} - 3 m^{2} + 3 m - 1 = 0$

Solving we get $m = 1, 1, 1$

Therefore, the complimentary solution is given by $y_{c} = c_{1} e^{x} + c_{2} x e^{x} + c_{3} x^{2} e^{x}$

Calculate Wornkians

Compare with. $C F = c_{1} y_{1} (x) + c_{2} y_{2} (x) + c_{3} y_{3} (x)$

$y_{1} (x) = e^{x}, y_{2} (x) = x e^{x} and y_{3} (x) = x^{2} e^{x}$

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

Calculate Wornkians

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

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

The particular solution is thus given by:

$y_{p} = e^{x} 鈭� \frac{x^{2} e^{2 x} (e^{x})}{2 e^{3 x}} d x + x e^{x} 鈭� \frac{- 2 x e^{2 x} (e^{x})}{2 e^{3 x}} d x + x^{2} e^{x} 鈭� \frac{e^{2 x} 路 e^{x}}{2 e^{3 x}} d x y_{p} = e^{x} \frac{x^{3}}{6} - \frac{x^{3}}{2} e^{x} + \frac{x^{3}}{2} e^{x} = \frac{x^{3} e^{x}}{6}$

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91影视

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} - 3 y^{''} + 3 y^{'} - y = e^{x}$

Short Answer

Step by step solution

Definition

Find complementary solution

Calculate Wornkians

Calculate Wornkians

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

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Probability and Statistics

Applied Mathematics

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91影视

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.y'''-3y''+3y'-y=ex

Short Answer

Step by step solution

Definition

Find complementary solution

Calculate Wornkians

Calculate Wornkians

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} - 3 y^{''} + 3 y^{'} - y = e^{x}$