Q5E find a general solution to the g... [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: Q5E (page 337)

find a general solution to the given equation.
$y''' - 2 y'' - 5 y' + 6 y = e^{x} + x^{2}$

Short Answer

Expert verified

$y (x) = c_{1} e^{- 2 x} + c_{2} e^{x} + c_{3} e^{3 x} - \frac{1}{6} x e^{x} + \frac{37}{108} + \frac{5 x}{18} + \frac{x^{2}}{6}$

Step by step solution

Find the corresponding auxiliaryequation

Theauxiliary equationof corresponding homogeneous equation

$r^{3} - 2 r^{2} - 5 r + 6 = (r - 1) (r - 3) (r + 2) = 0$

The solutions of the auxiliary equation are

$r = - 2, r = 1, r = 3$

Therefore a general solution to the homogeneous equation is

$y_{h} (x) = c_{1} e^{- 2 x} + c_{2} e^{x} + c_{3} e^{3 x}$

Find particular solution

Let the particular solution be

$y_{p} (x) = a x e^{x} + b + c x + d x^{2}$

Then

$y_{p}^{'} (x) = a e^{x} + a x e^{x} + c + 2 d x y_{p}^{''} (x) = 2 a e^{x} + a x e^{x} + c + 2 d y_{p}^{'''} (x) = 3 a e^{x} + a x e^{x}$

Then

$y_{p}^{'''} (x) - 2 y_{p}^{''} (x) - 5 y_{p}^{'} (t) + 6 y_{p} (x) = 3 a e^{x} + a x e^{x} - 4 a e^{x} - 2 a x e^{x} - 4 d - 5 a e^{x} - 5 a x e^{x} - 5 c - 10 d x + 6 a x e^{x} + 6 b + 6 c x + 6 d x^{2} = - 6 a e^{x} + (6 b - 5 c - 4 d) + (6 c - 10 d) x + 6 d x^{2}$

If $- 6 a e^{x} + (6 b - 5 c - 4 d) + (6 c - 10 d) x + 6 d x^{2} = e^{x} + x^{2}$

Then $- 6 a = 1, 6 b - 5 c - 4 d = 0, 6 c - 10 d = 0$

Then $a = - \frac{1}{6}, b = \frac{37}{108}, c = \frac{5}{18}, d = \frac{1}{6}$

Hence $y_{p} (x) = - \frac{1}{6} x e^{x} + \frac{37}{108} + \frac{5 x}{18} + \frac{x^{2}}{6}$

y(x)=yh+yp

Then $y (x) = c_{1} e^{- 2 t} + c_{2} e^{t} + c_{3} e^{3 t} - \frac{1}{6} x e^{x} + \frac{37}{108} + \frac{5 x}{18} + \frac{x^{2}}{6}$

Is the general solution of $y''' - 2 y'' - 5 y' + 6 y = e^{x} + x^{2}$

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91影视

find a general solution to the given equation.
$y''' - 2 y'' - 5 y' + 6 y = e^{x} + x^{2}$

Short Answer

Step by step solution

Find the corresponding auxiliaryequation

Find particular solution

y(x)=yh+yp

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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91影视

find a general solution to the given equation.y'''-2y''-5y'+6y=ex+x2

Short Answer

Step by step solution

Find the corresponding auxiliaryequation

Find particular solution

y(x)=yh+yp

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Decision Maths

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

find a general solution to the given equation.
$y''' - 2 y'' - 5 y' + 6 y = e^{x} + x^{2}$