Q21 E Determine which values of m the ... [FREE SOLUTION]

91影视

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 1: Q21 E (page 1)

Determine which values of m the function $蠒 (x) = x^{m}$ is a solution to the given equation.
(a) $3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0$
(b) $x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0$

Short Answer

Expert verified

$m = \frac{1}{3}, - 3$
$m = 1 卤 \sqrt{6}$

Step by step solution

(a): Taking the given function as y

First of all, we will take $蠒 (x) = y$

Differentiating the given function

Differentiatingconcerning x,

$蠒' (x) = \frac{dy}{dx} = m x^{m - 1}$

Again, differentiating concerning x,

$蠒'' (x) = \frac{d^{2} y}{{dx}^{2}} = m (m - 1) x^{m - 2}$

Substituting the values from step 2 in the given differential equation

$\begin{array}{l} 3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0 \\ 3 x^{2} [m (m - 1) x^{m - 2}] + 11 (x) {mx}^{m - 1} - 3 x^{m} = 0 \\ 3 m^{2} x^{m} - 3 {mx}^{m} + 11 {mx}^{m} - 3 x^{m} = 0 \\ 3 m^{2} + 8 m - 3 = 0 \\ (m - \frac{1}{3}) (m + 3) = 0 \\ m = \frac{1}{3}, - 3 \end{array}$

Hence, the values of m are $\frac{1}{3}$ and -3.

Step 4(b): Taking the given function as y

First of all, we will take $蠒 (x) = y$ .

Differentiating the given function

Differentiatingconcerning x,

$蠒' (x) = \frac{dy}{dx} = m x^{m - 1}$

Again, differentiating concerning x,

$蠒'' (x) = \frac{d^{2} y}{{dx}^{2}} = m (m - 1) x^{m - 2}$

Substituting the values from step 2 in the given differential equation.

$\begin{array}{l} x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0 \\ x^{2} m (m - 1) x^{m - 2} - {xmx}^{m - 1} - 5 x^{m} = 0 \\ m^{2} x^{m} - {mx}^{m} - {mx}^{m} - 5 x^{m} = 0 \\ m^{2} x^{m} - 2 {mx}^{m} - 5 x^{m} = 0 \\ (m^{2} - 2 m - 5) x^{m} = 0 \\ m^{2} - 2 m - 5 = 0 \\ m = 1 卤 \sqrt{6} \end{array}$

Hence, the values of m are $(1 + \sqrt{6})$ and $(1 - \sqrt{6})$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Determine which values of m the function $蠒 (x) = x^{m}$ is a solution to the given equation.
(a) $3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0$
(b) $x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0$

Short Answer

Step by step solution

(a): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation

Step 4(b): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Logic and Functions

Geometry

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Determine which values of m the function蠒(x)=xmis a solution to the given equation.(a)3x2d2ydx2+11xdydx-3y=0(b)x2d2ydx2-xdydx-5y=0

Short Answer

Step by step solution

(a): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation

Step 4(b): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Logic and Functions

Geometry

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Determine which values of m the function $蠒 (x) = x^{m}$ is a solution to the given equation.
(a) $3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0$
(b) $x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0$