Q41E Find a second linearly independe... [FREE SOLUTION]

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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: Q41E (page 200)

Find a second linearly independent solution using reduction of order.
$t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ .

Short Answer

Expert verified

The second linearly independent solution of the given equation $t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ is $y = \frac{1}{5 t^{2}} - \frac{1}{20} + c_{1} t^{- 1} + c_{2} t^{4} lnt$ .

Step by step solution

Finding a homogeneous solution

Given differential equation is $t^{2} y'' - 2 ty' - 4 y = t^{- 1}$

Let $y = t^{r}$ and then find the solution to the associated homogeneous function;

$y' (t) = r t^{r - 1} y'' (t) = r (r - 1) t^{r - 2}$

Substitute these in the differential equation:

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

$r = - 1$ and $r = 4$

So, the homogenous solution is $y = c_{1} t^{- 1} + c_{2} t^{4}$

Finding v1,v2

Now find the non-homogenous solution by using the variation of parameter method:

$\begin{array}{rcl} aW [y_{1}, y_{2}] & = & t^{2} ((t^{- 1} (4 t^{3}) - (- t^{- 2} (t^{4})) \\ = & t^{2} (5 t^{2}) \\ = & 5 t^{4} \end{array}$

And

$\begin{array}{rcl} v_{1} & = & 鈭� - \frac{f (t) y_{2} (t)}{aW [y_{1}, y_{2}]} dt \\ = & 鈭� - \frac{t^{- 2} t^{4}}{5 t^{4}} dt \\ = & \frac{1}{5 t} \end{array}$

And

$\begin{array}{rcl} v_{2} & = & 鈭� \frac{f (t) y_{1} (t)}{aW [y_{1}, y_{2}]} dt \\ = & 鈭� \frac{t^{- 2} t^{- 1}}{5 t^{2}} dt \\ = & - \frac{1}{20 t^{4}} \end{array}$

Hence,

$\begin{array}{rcl} y_{p} & = & \frac{1}{5 t} (t^{- 1}) - \frac{1}{20 t^{4}} (t^{4}) \\ = & \frac{1}{5 t^{2}} - \frac{1}{20} \end{array}$

Therefore, the total solution is $y = \frac{1}{5 t^{2}} - \frac{1}{20} + c_{1} t^{- 1} + c_{2} t^{4} lnt$

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91影视

Find a second linearly independent solution using reduction of order.
$t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ .

Short Answer

Step by step solution

Finding a homogeneous solution

Finding v1,v2

One App. One Place for Learning.

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91影视

Find a second linearly independent solution using reduction of order.t2y''-2ty'-4y=0,t>0;f(t)=t-1.

Short Answer

Step by step solution

Finding a homogeneous solution

Finding v1,v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Pure Maths

Probability and Statistics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Find a second linearly independent solution using reduction of order.
$t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ .