Q23E In Problems 22 through 25, use a... [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: Q23E (page 192)

In Problems 22 through 25, use a variation of parameters to find a general solution to the differential equation given that the functions $y_{1}$ and $y_{2}$ are linearly independent solutions to the corresponding homogeneous equation for t> 0.
$ty'' - (t + 1) y' + y = t^{2}; y_{1} = e^{t}, y_{2} = t + 1$

Short Answer

Expert verified

The general solution is $y (t) = c_{1} t^{2} + c_{2} t^{3} + (- t + \frac{t^{- 2}}{2}) t^{2} + (\ln | t | - \frac{t^{- 3}}{3}) t^{3}$ .

Step by step solution

Find a particular solution.

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

And $y_{1} = e^{t}, y_{2} = t + 1$

$y_{h} (t) = c_{1} e^{t} + c_{2} (t + 1)$

The particular solution is $y_{p} = v_{1} (t) e^{t} + v_{2} (t) (t + 1) .$

Evaluate v1 and v2.

Here $y_{p} = v_{1} (t) e^{t} + v_{2} (t) (t + 1)$

$\begin{matrix} v_{1}' = \frac{- f (t) y_{2} (t)}{a [y_{1} (t) y'_{2} (t) - y'_{1} (t) y_{2} (t)]} \\ = \frac{- t^{2} (t + 1)}{t [e^{t} . 1 - e^{t} (t + 1)]} \\ = \frac{(t + 1)}{[e^{t}]} \end{matrix}$

Now integrate the above result.

$\begin{matrix} v_{1} (t) = 鈭� \frac{(t + 1)}{[e^{t}]} dt \\ = - (t + 2) e^{- 1} \end{matrix}$

Determine v'2 and v2

$\begin{matrix} v_{2}' = \frac{f (t) y_{1} (t)}{a [y_{1} (t) y'_{2} (t) - y'_{1} (t) y_{2} (t)]} \\ = \frac{- t^{2} e^{t}}{t [e^{t} . 1 - e^{t} (t + 1)]} \\ = - 1 \end{matrix}$

Integrate the above result.

$\begin{array}{l} v_{2} (t) = 鈭� 鈭� 1 dt \\ = - t \end{array}$

Thus, a particular solution is:

$\begin{array}{l} y_{p} = (- (t + 2) e^{- 1}) e^{t} - t (t + 1) \\ y_{p} = - t^{2} - 2 t - 2 \end{array}$

And the general solution is:

$\begin{matrix} y (t) = y_{h} (t) + y_{p} (t) \\ y (t) = c_{1} e^{t} + c_{2} (t + 1) - t^{2} - 2 t - 2 \end{matrix}$

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91影视

Short Answer

Step by step solution

Find a particular solution.

Evaluate v1 and v2.

Determine v'2 and v2

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91影视

In Problems 22 through 25, use a variation of parameters to find a general solution to the differential equation given that the functions y1 and y2are linearly independent solutions to the corresponding homogeneous equation for t> 0.ty''-(t+1)y'+y=t2;y1=et,y2=t+1

Short Answer

Step by step solution

Find a particular solution.

Evaluate v1 and v2.

Determine v'2 and v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Decision Maths

Probability and Statistics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.