Chapter 6: Q7E (page 337)

find a general solution to the given equation.
$y''' + 3 y'' - 4 y = e^{- 2 x}$

Short Answer

Expert verified

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

Step by step solution

Find the corresponding auxiliaryequation

Theauxiliary equationof corresponding homogeneous equation

$r^{3} + 3 r^{2} - 4 = (r - 1) (r + 2)^{2} = 0$

The solutions of the auxiliary equation are

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

Therefore a general solution to the homogeneous equation is

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

Find particular solution

Let the particular solution be

$y_{p} (x) = a x^{2} e^{- 2 x}$

Then

$y_{p}^{'} (x) = 2 a x e^{- 2 x} - 2 a x^{2} e^{- 2 x} y_{p}^{''} (x) = 2 a e^{- 2 x} - 8 a x e^{- 2 x} + 4 a x^{2} e^{- 2 x} y_{p}^{'''} (x) = - 12 a e^{- 2 x} + 24 a x e^{- 2 x} - 8 a x^{2} e^{- 2 x}$

Then

$y_{p}^{'''} (x) + 3 y_{p}^{''} (x) - 4 y_{p} (x) = - 12 a e^{- 2 x} + 24 a x e^{- 2 x} - 8 a x^{2} e^{- 2 x} + 6 a e^{- 2 x} - 24 a x e^{- 2 x} + 12 a x^{2} e^{- 2 x} - 4 a x^{2} e^{- 2 x} = - 6 a e^{- 2 x} .$

If $- 6 a e^{- 2 x} = e^{- 2 x} t h e n 6 a = 1$

Then $a = - \frac{1}{6}$

Hence

$y_{p} (x) = - \frac{1}{6} x^{2} e^{- 2 x}$

Step 3: y(x)=yh+yp

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

Is the general solution of $y''' + 3 y'' - 4 y = e^{- 2 x}$

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find a general solution to the given equation.
$y''' + 3 y'' - 4 y = e^{- 2 x}$

Short Answer

Step by step solution

Find the corresponding auxiliaryequation

Find particular solution

Step 3: y(x)=yh+yp

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Most popular questions from this chapter

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find a general solution to the given equation.y'''+3y''-4y=e-2x

Short Answer

Step by step solution

Find the corresponding auxiliaryequation

Find particular solution

Step 3: y(x)=yh+yp

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

find a general solution to the given equation.
$y''' + 3 y'' - 4 y = e^{- 2 x}$