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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 6: Q4E (page 332)

Find a general solution for the differential equation with x as the independent variable: $y''' - 3 y'' - y' + 3 y = 0$

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variableis. $y (x) = c_{1} e^{- x} + c_{2} e^{- 5 x} + c_{3} e^{4 x}$

Step by step solution

01

Auxiliary equation:

The auxiliary equation is $r^{3} + 2 r^{2} - 19 r - 20 = 0$

02

Inspecting the sum further:

By inspection, we find that r = -1 is a root and using polynomial division we get

$\begin{array}{l} r^{3} + 2 r^{2} - 19 r - 20 = (r + 1) (r^{2} + r - 20) \\ = (r + 1) (r + 5) (r - 4) \\ = 0 \end{array}$

03

General solution:

Now the roots of auxiliary equation are $r_{1} = - 1, r_{2} = - 5$ and . $r_{3} = 4$ Therefore, a general solution to given equation is

$y (x) = c_{1} e^{- x} + c_{2} e^{- 5 x} + c_{3} e^{4 x}$

Hence the final solution is $y (x) = c_{1} e^{- x} + c_{2} e^{- 5 x} + c_{3} e^{4 x}$

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Most popular questions from this chapter

Use the reduction of order method described in Problem 31 to find three linearly independent solutions to, given $y''' - 2 y'' + y' - 2 y = 0$ that $f (x) = e^{2 x}$ is a solution.

As an alternative proof that the functions $e^{r_{1} x}, e^{r_{2} x}, e^{r_{3} x}, ..., e^{r_{n} x}$ are linearly independent on (∞,-∞) when $r_{1}, r_{2}, . .. r_{n}$ are distinct, assume $C_{1} e^{r_{1} x} + C_{2} e^{r_{2} x} + C_{3} e^{r_{3} x} + . .. + C_{n} e^{r_{n} x}$ holds for all x in (∞,-∞) and proceed as follows:

(a) Because the ri’s are distinct we can (if necessary)relabel them so that $r_{1} > r_{2} > r_{3} > . .. > r_{n}$ .Divide equation (33) by to obtain $C_{1} + \frac{C_{2} e^{r_{2} x}}{e^{r_{2} x}} + \frac{C_{3} e^{r_{3} x}}{e^{r_{3} x}} + . .. + \frac{C_{n} e^{r_{n} x}}{e^{r_{n} x}} = 0$ Now let x→∞ on the left-hand side to obtainC1 = 0.(b) Since C1 = 0, equation (33) becomes

$C_{2} e^{r_{2} x} + C_{3} e^{r_{3} x} + . .. + C_{n} e^{r_{n} x}$ = 0for all x in(∞,-∞). Divide this equation by $e^{r_{2} x}$

and let x→∞ to conclude that C2 = 0.

(c) Continuing in the manner of (b), argue that all thecoefficients, C1, C2, . . . ,Cn are zero and hence $e^{r_{1} x}, e^{r_{2} x}, e^{r_{3} x}, . .., e^{r_{n} x}$ are linearly independent on(∞,-∞).

In Problems 38 and 39, use the elimination method of Sectionto find a general solution to the given system.

$d^{2} x / d t^{2} - x + y = 0 x + d^{2} y / d t^{2} - y = e^{3 t}$

Let y₁x₂= Ce^rx, where C (≠0) and r are real numbers,be a solution to a differential equation. Supposewe cannot determine r exactly but can only approximateit by $r$ . Let (x) =Ce^rxand consider the error

$| y (x) - \tilde{y} (x) |$

(a) If r and $\tilde{r}$ are positive, r ≠ , show that the errorgrows exponentially large as x approaches + ∞.

(b) If r and $r$ are negative, r≠ , show that the errorgoes to zero exponentially as x approaches + ∞.

Determine whether the given functions are linearly dependent or linearly independent on the interval $(0, \infty)$ .

(a) ${e^{2 x}, x^{2} e^{2 x}, e^{- x}}$

(b) ${e^{x} s i n 2 x, x e^{x} s i n 2 x, e^{x}, x e^{x}}$

(c) ${2 e^{2 x} - e^{x}, e^{2 x} + 1, e^{2 x} - 3, e^{x} + 1}$

See all solutions

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