Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.

6. $\frac{d x}{d t} = \frac{1}{x}, x (0) = 1$

The solution is $x (t) = \sqrt{2 t + 1}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Simplification for the differential equation

Consider the equation as follows:

$\frac{d x}{d t} = \frac{1}{x}$

Now, separate the variables as follows:

$\begin{matrix} \frac{d x}{d t} = \frac{1}{x} \\ x d x = d t \end{matrix}$

Integrating on both sides as follows:

$\begin{matrix} x d x = d t \\ \int x d x = \int d t \\ \frac{x^{2}}{2} = t + C \end{matrix}$

Substituting the initial condition as follows:

$\begin{matrix} \frac{x^{2}}{2} = t + C \\ \frac{{(1)}^{2}}{2} = 0 + C {Q x (0) = 1} \\ \frac{1}{2} = C \end{matrix}$

Step 2 : Calculation of the solution

Now, substitute the value $\frac{1}{2}$ for C in $\frac{x^{2}}{2} = t + C$ as follows:

$\begin{matrix} \frac{x^{2}}{2} = t + C \\ \frac{x^{2}}{2} = t + \frac{1}{2} \\ \frac{x^{2}}{2} = \frac{2 t}{2} + \frac{1}{2} \\ \frac{x^{2}}{2} = \frac{2 t + 1}{2} \end{matrix}$

Simplify further as follows:

$\begin{matrix} \frac{x^{2}}{2} = \frac{2 t + 1}{2} \\ x^{2} = 2 t + 1 \\ x = \sqrt{2 t + 1} \\ x (t) = \sqrt{2 t + 1} \end{matrix}$

Hence, the solution for the differential equation $\frac{d x}{d t} = \frac{1}{x}$ is $x (t) = \sqrt{2 t + 1}$

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.

6. $\frac{d x}{d t} = \frac{1}{x}, x (0) = 1$

Step by Step Solution

Step 1: Simplification for the differential equation

Step 2 : Calculation of the solution

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Linear Algebra With Applications

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Short Answer

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxasdxfx=dt and integrate both sides. 6.dxdt=1x,x(0)=1

Step by Step Solution

Step 1: Simplification for the differential equation

Step 2 : Calculation of the solution

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Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.

6. $\frac{d x}{d t} = \frac{1}{x}, x (0) = 1$