Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Solve the differential equation $\frac{d x}{d t} + 3 x = 7$ and find the solution of the differential equation.

The solution is $f (t) = \frac{7}{3} + \frac{7}{3} e^{- 3 t} C$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Definition of first order linear differential equation

Consider the differential equation $f' (t) - a f (t) = g (t)$ where $g (t)$ is a smooth function and $' a'$ is a constant. Then the general solution will be $f (t) = e^{a t} \int e^{- a t} g (t) d t$ .

Step2: Determination of the solution

Consider the differential equation as follows.

$x' (t) + 3 x (t) = 7$

Now, the differential equation is in the form as follows.

$f' (t) - a f (t) = g (t)$ , where $g (t)$ is a smooth function, then the general solution will be as follows.

$f (t) = e^{a t} \int e^{- a t} g (t) d t$

Step3: Compute the calculation of the solution

Substitute the value $7$ for $g (t)$ and $- 3$ for in $f (t) = e^{a t} \int e^{- a t} g (t) d t$ as follows.

$\begin{array}{l} f (t) = e^{a t} \int e^{- a t} g (t) d t A \\ f (t) = e^{- 3 t} \int e^{3 t} \times 7 \times d t \\ f (t) = 7 e^{- 3 t} \int e^{3 t} d t \end{array}$

Using substitution method in the integral as follows.

$\begin{matrix} y = 3 t \\ d y = 3 d t \\ \frac{d y}{3} = d t \end{matrix}$

Substitute the value $3 t$ for $y$ and $\frac{d y}{3}$ for $d t$ in $f (t) = 7 e^{3 t} \int e^{- 3 t} d t$ as follows.

$\begin{array}{l} f (t) = 7 e^{- 3 t} \int e^{3 t} d t \\ f (t) = 7 e^{- 3 t} \int e^{y} \frac{d y}{3} \\ f (t) = \frac{7}{3} e^{- 3 t} (e^{y} + C) \end{array}$

Now, undo the substitution as follows.

$\begin{matrix} f (t) = \frac{7}{3} e^{- 3 t} (e^{y} + C) \\ f (t) = \frac{7}{3} e^{- 3 t} (e^{3 t} + C) \\ = \frac{7}{3} e^{- 3 t} e^{3 t} + \frac{7}{3} e^{- 3 t} C \\ f (t) = \frac{7}{3} + \frac{7}{3} e^{- 3 t} C \end{matrix}$

Hence, the solution for the linear differential equation $x' (t) + 3 x (t) = 7$ is $f (t) = \frac{7}{3} + \frac{7}{3} e^{- 3 t} C$

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Solve the differential equation $\frac{d x}{d t} + 3 x = 7$ and find the solution of the differential equation.

Step by Step Solution

Step1: Definition of first order linear differential equation

Step2: Determination of the solution

Step3: Compute the calculation of the solution

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Solve the differential equation dxdt+3x=7 and find the solution of the differential equation.

Step by Step Solution

Step1: Definition of first order linear differential equation

Step2: Determination of the solution

Step3: Compute the calculation of the solution

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Solve the differential equation $\frac{d x}{d t} + 3 x = 7$ and find the solution of the differential equation.