Chapter 4: Q4E (page 191)

In Problems 1–8, 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) $y (t) = c_{1} e^{α t} \cos β t c_{2} e^{α t} \sin β t,$ 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;

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

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:

$v_{2} (t) = \int 1 d t = t + C$

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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In Problems 1–8, 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

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In Problems 1–8, 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

Applied Mathematics

Mechanics Maths

Pure Maths

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

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