Q 2.6-41E Question: Use the substitution v... [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 2: Q 2.6-41E (page 77)

Question: Use the substitution $v = x - y + 2$ to solve equation (8).
$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

Short Answer

Expert verified

${(x - y + 2)}^{2} = C e^{2 x} + 1$

Step by step solution

Use the given substitution v=x-y+2 to solve the given equation,

The given substitution,

$v = x - y + 2$

Simplify the above equation,

$\begin{array}{l} 1 - v = 1 - (x - y + 2) \\ 1 - v = y - x - 1 \end{array}$

And

role="math" localid="1663933729097" $\begin{matrix} 1 - v = y - x - 1 \\ y = 1 - v + x + 1 \\ y = x - v + 2 \\ \frac{dy}{dx} = 1 - \frac{dv}{dx} \end{matrix}$

Given equation,

$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

Substitute $v = x - y + 2, 鈥� 1 - v = y - x - 1$ and $\frac{dy}{dx} = 1 - \frac{dv}{dx}$ in the above equation,

$1 - \frac{dv}{dx} = 1 - v + {(v)}^{- 1}$

Simplify the above equation,

$\begin{array}{l} - \frac{dv}{dx} = - v + {(v)}^{- 1} \\ \frac{dv}{dx} = v - \frac{1}{v} \\ \frac{dv}{dx} = \frac{v^{2} - 1}{v} \end{array}$

Cross multiplication on both sides in the above equation,

$\begin{array}{l} \frac{v}{v^{2} - 1} dv = dx \\ \frac{2 v}{v^{2} - 1} dv = 2 鈥� dx \end{array}$

Step 2: Integrating

Taking integration both sides in the above equation

$鈭� \frac{2 v}{v^{2} - 1} dv = 2 鈭� dx$

Solve the above equation,

$\begin{matrix} 鈭� \frac{2 v}{v^{2} - 1} dv = 2 鈭� dx \\ \log (v^{2} - 1) = 2 x + \log 鈥� C \\ \frac{v^{2} - 1}{C} = e^{2 x} \\ v^{2} - 1 = C 鈥� e^{2 x} \\ v^{2} = C 鈥� e^{2 x} + 1 \end{matrix}$

Substitute the value of $v = x - y + 2$ in the above equation,

${(x - y + 2)}^{2} = C e^{2 x} + 1$

Hence the solution is ${(x - y + 2)}^{2} = C e^{2 x} + 1$

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

Question: Use the substitution $v = x - y + 2$ to solve equation (8).
$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

Short Answer

Step by step solution

Use the given substitution v=x-y+2 to solve the given equation,

Step 2: Integrating

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Mechanics Maths

Calculus

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

Question: Use the substitution v=x-y+2 to solve equation (8).dydx=y-x-1+x-y+2-1

Short Answer

Step by step solution

Use the given substitution v=x-y+2 to solve the given equation,

Step 2: Integrating

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Mechanics Maths

Calculus

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Question: Use the substitution $v = x - y + 2$ to solve equation (8).
$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$