Q43E Find a second linearly independe... [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 4: Q43E (page 200)

Find a second linearly independent solution using reduction of order.
$tx'' - (t + 1) x' + x = 0, t > 0; f (t) = e^{t}$

Short Answer

Expert verified

The second linearly independent solution of the given equation

$tx'' - (t + 1) x' + x = 0, t > 0; f (t) = e^{t}$ is $y = c_{1} e^{t} - c_{2} \frac{t + 1}{e^{2 t}}$

Step by step solution

Finding y2

Given differential equation is $tx'' - (t + 1) x' + x = 0$ then the standard form of the solution is $x'' - \frac{t + 1}{t} x' + \frac{1}{t} x = 0$ and $f (t) = e^{t}$ is one of the solutions and role="math" localid="1664197935206" $p (t) = - \frac{t + 1}{t}$ . Then;

$\begin{array}{rcl} y_{2} & = & y_{1} (t) 鈭� \frac{e^{- 鈭� p (t)} dt}{y_{1}^{2} (t)} dt \\ = & e^{t} 鈭� \frac{e^{鈭� \frac{1 + t}{t}} dt}{{(e^{t})}^{2}} dt \\ = & e^{t} 鈭� \frac{e^{\ln (t^{1}) + t}}{e^{2 t}} \end{array}$

Simplification

Simplify the above

$= e^{t} 脳鈭� \frac{t 脳 e^{t}}{e^{2 t}} = e^{t} 脳 \frac{(- t - 1) e^{- t}}{e^{2 t}} = - \frac{t + 1}{e^{2 t}}$

So, the solution is $y = c_{1} e^{t} - c_{2} \frac{t + 1}{e^{2 t}}$ .

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

Find a second linearly independent solution using reduction of order.
$tx'' - (t + 1) x' + x = 0, t > 0; f (t) = e^{t}$

Short Answer

Step by step solution

Finding y2

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Geometry

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

Find a second linearly independent solution using reduction of order.tx''-(t+1)x'+x=0,t>0;f(t)=et

Short Answer

Step by step solution

Finding y2

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Geometry

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a second linearly independent solution using reduction of order.
$tx'' - (t + 1) x' + x = 0, t > 0; f (t) = e^{t}$