Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Use the method discussed under “Bernoulli Equations” to solve problems 21-28.
$\frac{dx}{dt} + {tx}^{3} + \frac{x}{t} = 0$

Equation of the form of Bernoulli equation for the given equation is $x^{- 2} = 2 t^{2} \ln |t| + {Ct}^{2}$ .

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Step by Step Solution

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General form of Bernoulli equation

Bernoulli’s equation

A first-order equation that can be written in the form $\frac{dy}{dx} + P (x) y = Q (x) y^{n}$ , where $P (x)$ and $Q (x)$ are continuous on an interval $(a, b)$ and is a real number, is called a Bernoulli equation.

Evaluate the given equation

Given, $\frac{dx}{dt} + {tx}^{3} + \frac{x}{t} = 0$ .

Compare with the general form of the Bernoulli equation.

$n = 3, P (t) = \frac{1}{t} and Q (t) = - t$

Now, divide by $x^{3}$ , we get,

$x^{- 3} \frac{dx}{dt} + \frac{x^{- 2}}{t} = - t \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Substitute $v = x^{- 2}$ .

Differentiate with respect to t.

$\begin{matrix} \frac{dv}{dt} = - 2 x^{- 3} \frac{dx}{dt} \\ - \frac{1}{2} \frac{dv}{dt} = x^{- 3} \frac{dx}{dt} \end{matrix}$

Substitute it on equation (1)

$\begin{matrix} - \frac{1}{2} \frac{dv}{dt} + \frac{v}{t} = - t \\ \frac{dv}{dt} - \frac{2}{t} v = 2 t \cdot \cdot \cdot \cdot \cdot \cdot (2) \end{matrix}$

Integrate the equation

Now, integrate $P (t)$ the first. Where role="math" localid="1663935034942" $P (x) = - \frac{2}{t}$ .

$\begin{matrix} \int P (t) dt = - 2 \int \frac{1}{t} dt \\ = - 2 \ln |t| \end{matrix}$

Then,

$\begin{matrix} μ (t) = e^{\int P (t) dt} \\ = e^{- 2 \ln |t|} \\ = t^{- 2} \end{matrix}$

Multiply $μ (t)$ with equation (2).

$\begin{matrix} t^{- 2} \frac{dv}{dt} - 2 t^{- 3} v = 2 t^{- 1} \\ \frac{d}{dt} [t^{- 2} v] = 2 t^{- 1} \end{matrix}$

Integrate both sides,

$\begin{matrix} \int \frac{d}{dt} [t^{- 2} v] dt = \int 2 t^{- 1} dt \\ t^{- 2} v = 2 \int \frac{1}{t} dt \\ = 2 \ln |t| + C_{1} \\ v = 2 t^{2} \ln |t| + {Ct}^{2} \end{matrix}$

Substitution method

Substitute $v = x^{- 2}$ .

$x^{- 2} = 2 t^{2} \ln |t| + {Ct}^{2}$

Hence, the solution is $x^{- 2} = 2 t^{2} \ln |t| + {Ct}^{2}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Use the method discussed under “Bernoulli Equations” to solve problems 21-28.
$\frac{dx}{dt} + {tx}^{3} + \frac{x}{t} = 0$

Step by Step Solution

General form of Bernoulli equation

Evaluate the given equation

Integrate the equation

Substitution method

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Use the method discussed under “Bernoulli Equations” to solve problems 21-28.dxdt+tx3+xt=0

Step by Step Solution

General form of Bernoulli equation

Evaluate the given equation

Integrate the equation

Substitution method

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Use the method discussed under “Bernoulli Equations” to solve problems 21-28.
$\frac{dx}{dt} + {tx}^{3} + \frac{x}{t} = 0$