Q38E Find general solutions to the no... [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: Q38E (page 200)

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.
$t^{2} y'' + 3 ty' + y = t^{- 1}$

Short Answer

Expert verified

The solution of the given equation $t^{2} y'' + 3 ty' + y = t^{- 1}$ is localid="1664191735862" $y = - \frac{t^{- 1} lnt}{2} + \frac{1}{4 t^{3}} + c_{1} t^{- 1} + c_{2} t^{- 1} lnt$ .

Step by step solution

Substitute the values

Given differential equation is $t^{2} y'' + 3 ty' + y = t^{- 1}$

Letand 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:

$\begin{array}{rcl} t^{2} (r (r - 1) t^{r - 2} + 3 t ({rt}^{r - 1})) + t^{r} & = & 0 \\ (r^{2} + 2 r + 1) t^{r} & = & 0 \\ r^{2} + 2 r + 1 & = & 0 \\ {(r + 1)}^{2} & = & 0 \\ r & = & - 1 \end{array}$

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

Finding v1

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

$\begin{array}{rcl} aW [y_{1}, y_{2}] & = & t^{2} ((t^{- 1} (- \frac{lnt - 1}{t^{2}}) - (- t^{- 2} (t^{- 1} lnt)) \\ = & t^{2} (\frac{1}{t^{3}}) 鈬� \frac{1}{t} \end{array}$

And

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

Finding v2

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

Therefore,

$\begin{array}{rcl} y_{p} & = & \frac{2 lnt + 1}{4 t^{2}} (t^{- 1}) - \frac{1}{t^{2}} (t^{- 1} lnt) \\ = & - \frac{t^{- 1} lnt}{2} + \frac{1}{4 t^{3}} \end{array}$

Therefore, the total solution is $y = - \frac{t^{- 1} lnt}{2} + \frac{1}{4 t^{3}} + c_{1} t^{- 1} + c_{2} t^{- 1} lnt$ .

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91影视

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.
$t^{2} y'' + 3 ty' + y = t^{- 1}$

Short Answer

Step by step solution

Substitute the values

Finding v1

Finding v2

One App. One Place for Learning.

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91影视

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.t2y''+3ty'+y=t-1

Short Answer

Step by step solution

Substitute the values

Finding v1

Finding v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Discrete Mathematics

Calculus

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.
$t^{2} y'' + 3 ty' + y = t^{- 1}$