Chapter 6: Q4E (page 337)

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.
$y''' + y'' - 2 y = x e^{x} + 1$

Short Answer

Expert verified

$y_{p} (x) = - \frac{4}{25} x e^{x} + \frac{1}{10} x^{2} e^{x} - \frac{1}{2}$

Step by step solution

Find the corresponding auxiliaryequation

Theauxiliary equationof corresponding homogeneous equation

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

The solutions of the auxiliary equation are

$r = - 1 + i, r = - 1 - i, r = 1$

Find particular solution

Let the particular solution be

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

Then

$y_{p}^{'} (x) = a e^{x} + (a + 2 b) x e^{x} + b x^{2} e^{x} y_{p}^{''} (x) = (2 a + 2 b) e^{x} + (a + 4 b) x e^{x} + b x^{2} e^{x} y_{p}^{'''} (x) = (3 a + 6 b) e^{x} + (a + 6 b) x e^{x} + b x^{2} e^{x}$

Then

$y_{p}^{'''} (x) + y_{p}^{''} (x) - 2 y_{p} (x) = (3 a + 6 b) e^{x} + (a + 6 b) x e^{x} + b x^{2} e^{x} + (2 a + 2 b) e^{x} + (a + 4 b) x e^{x} + b x^{2} e^{x} - 2 a x e^{x} - 2 b x^{2} e^{x} - 2 c = (5 a + 8 b) e^{x} + 10 b x e^{x} - 2 c$

If $(5 a + 8 b) e^{x} + 10 b x e^{x} - 2 c = x e^{x} + 1$

Then $5 a + 8 b = 0, 10 b = 1 a n d - 2 c = 1$

Then $a = - \frac{4}{25}, b = \frac{1}{10} a n d c = - \frac{1}{2}$

Hence $y_{p} (x) = - \frac{4}{25} x e^{x} + \frac{1}{10} x^{2} e^{x} - \frac{1}{2}$

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use the method of undetermined coefficients to determine the form of a particular solution for the given equation.
$y''' + y'' - 2 y = x e^{x} + 1$

Short Answer

Step by step solution

Find the corresponding auxiliaryequation

Find particular solution

One App. One Place for Learning.

Most popular questions from this chapter

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use the method of undetermined coefficients to determine the form of a particular solution for the given equation.y'''+y''-2y=xex+1

Short Answer

Step by step solution

Find the corresponding auxiliaryequation

Find particular solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Calculus

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.
$y''' + y'' - 2 y = x e^{x} + 1$