Q2.3-7E Question: In Problems 7-16, obta... [FREE SOLUTION]

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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 2: Q2.3-7E (page 54)

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} - y - e^{3 x} = 0$

Short Answer

Expert verified

The general solution to the given equation is $y = \frac{e^{3 x}}{2} + {Ce}^{x}$ .

Step by step solution

Method for solving linear equations

Write the equation in the standard form $\frac{dy}{dx} + P (x) y = Q (x)$ .
Calculate the integrating factorlocalid="1664262698741" role="math" $渭 (x)$ by the formula $渭 (x) = \exp [{鈭�}_{}^{} P (x) dx]$ .
Multiply the equation in standard form by $渭 (x)$ and, recalling that the left-hand side is just $\frac{d}{dx} [渭 (x) y]$ , obtain

$渭 (x) \frac{dy}{dx} + P 渭 (x) y = 渭 (x) Q (x) \frac{d}{dx} [渭 (x) y] = 渭 (x) Q (x)$

Integrate the last equation and solve for y by dividing by $渭 (x)$ to obtain $y (x) = \frac{1}{蟺 (x)} [{鈭�}_{}^{} 蟺 (x) Q (x) + c]$ . Here C is an arbitrary constant.

Solving the given equation

Given that,
$\frac{dy}{dx} - y - e^{3 x} = 0 . . . . . . . (1)$

Write the equation (1) into standard form of linear equation.

$\frac{dy}{dx} - y - e^{3 x} = 0 \frac{dy}{dx} - y = e^{3 x} . . . . . . . . . (2)$

Calculate the integrating factor of $渭 (x)$ .

Where $P (x) = - 1$

Then,

$\begin{array}{rcl} 渭 (x) & = & e^{{鈭�}_{}^{} P (x) dx} \\ = & e^{{鈭�}_{}^{} - 1 dx} \\ = & e^{- x} \end{array}$

Simplification method

Multiply $渭 (x)$ in equation (2)

$\begin{array}{rcl} e^{- x} \frac{dy}{dx} - e^{- x} y & = & e^{2 x} \\ \frac{d}{dx} [e^{- x} y] & = & e^{2 x} \end{array}$

Integrating both sides.

$\begin{array}{rcl} {鈭�}_{}^{} \frac{d}{dx} [e^{- x} y] d x & = & {鈭�}_{}^{} e^{2 x} dx \\ e^{- x} y & = & \frac{e^{2 x}}{2} + C \\ y & = & \frac{e^{3 x}}{2} + {Ce}^{x} \end{array}$

So, the solution is $\begin{array}{rcl} y & = & \frac{e^{3 x}}{2} + {Ce}^{x} \end{array}$ .

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91影视

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} - y - e^{3 x} = 0$

Short Answer

Step by step solution

Method for solving linear equations

Solving the given equation

Simplification method

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Most popular questions from this chapter

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91影视

Question: In Problems 7-16, obtain the general solution to the equation.dydx-y-e3x=0

Short Answer

Step by step solution

Method for solving linear equations

Solving the given equation

Simplification method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Pure Maths

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} - y - e^{3 x} = 0$