Q4E In Problems 1鈥�8, find a genera... [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: Q4E (page 191)

In Problems 1鈥�8, find a general solution to the differential equation using the method of variation of parameters.
$y'' + 2 y' + y = e^{- t}$

Short Answer

Expert verified

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

Step by step solution

Find the particular solution.

The homogenous equation is $r^{2} + 2 r + 1 = 0$ .

Two independent solutions are $r = - 1, - 1$ .

Then $y_{1} = e^{- t}, y_{2} = t e^{- t}$

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

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

Evaluate v1 and v2

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

And referring to (9) $y (t) = c_{1} e^{伪 t} \cos 尾 t c_{2} e^{伪 t} \sin 尾 t,$ and solve the system then put the value of $y_{p} = 0$ .

$v_{1}' e^{- t} + v_{2}' t e^{- t} = 0 v_{1}' e^{- t} + v_{2}' [e^{- t} - t e^{- t}] = \frac{f}{a} v_{1}' e^{- t} + v_{2}' [e^{- t} - t e^{- t}] = e^{- t} v_{1}' + v_{2}' [1 - t] = 1$

Find v'1 and v1

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

Now integrating this;

$v_{1} (t) = 鈭� - t 鈥� d t = - \frac{t^{2}}{2} + C$

Determine v'2 and v2

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

Integrate this:

$v_{2} (t) = 鈭� 1 鈥� d t = t + C$

Therefore, the particular solution is:

$y_{p} = (- \frac{t^{2}}{2} + C) e^{- t} + (t + C) t e^{- t} y_{p} = - \frac{t^{2}}{2} e^{- t}$

Hence, the general solution is:

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

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91影视

In Problems 1鈥�8, find a general solution to the differential equation using the method of variation of parameters.
$y'' + 2 y' + y = e^{- t}$

Short Answer

Step by step solution

Find the particular solution.

Evaluate v1 and v2

Find v'1 and v1

Determine v'2 and v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

In Problems 1鈥�8, find a general solution to the differential equation using the method of variation of parameters.y''+2y'+y=e-t

Short Answer

Step by step solution

Find the particular solution.

Evaluate v1 and v2

Find v'1 and v1

Determine v'2 and v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Pure Maths

Probability and Statistics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 1鈥�8, find a general solution to the differential equation using the method of variation of parameters.
$y'' + 2 y' + y = e^{- t}$