Q2.3-9E Question: In Problems 7-16, obta... [FREE SOLUTION]

91影视

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: Q2.3-9E (page 54)

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dr}{诲胃} + 迟补苍胃 = 蝉别肠胃$

Short Answer

Expert verified

The general solution to the given equation is $r = 蝉颈苍胃 + 颁肠辞蝉胃$ .

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 factor by $渭 (x)$ the formula .
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.

Step 2: Solve the given equation

Given that,

$\frac{dr}{诲胃} + 迟补苍胃 = 蝉别肠胃$

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

Where $P (胃) = tan 胃$ .

Then,

$\begin{array}{rcl} 渭 (x) & = & e^{{鈭�}_{}^{} P (胃) d 胃} \\ = & e^{{鈭�}_{}^{} 迟补苍胃诲胃} \\ = & e^{\ln |蝉别肠胃|} \\ = & 蝉别肠胃 \end{array}$

Simplification method

Multiply $渭 (x)$ in equation (2)

$\begin{array}{rcl} 蝉别肠胃 \frac{dr}{诲胃} + r蝉别肠胃迟补苍胃 & = & \sec^{2} 胃 \\ \frac{d}{诲胃} [r蝉别肠胃] & = & \sec^{2} 胃 \end{array}$

Integrating both sides.

role="math" localid="1664197341858" $\begin{array}{rcl} {鈭�}_{}^{} \frac{d}{诲胃} [r蝉别肠胃] & = & {鈭�}_{}^{} \sec^{2} 胃 \\ r蝉别肠胃 & = & 迟补苍胃 + C \\ r & = & \frac{迟补苍胃}{蝉别肠胃} + \frac{C}{蝉别肠胃} \\ r & = & 蝉颈苍胃 + 颁肠辞蝉胃 \end{array}$

So, the solution is

\begin{array}{rcl} r & = & 蝉颈苍胃 + 颁肠辞蝉胃 \end{array}

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91影视

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dr}{诲胃} + 迟补苍胃 = 蝉别肠胃$

Short Answer

Step by step solution

Method for solving linear equations

Step 2: Solve the given equation

Simplification method

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91影视

Question: In Problems 7-16, obtain the general solution to the equation.dr诲胃+迟补苍胃=蝉别肠胃

Short Answer

Step by step solution

Method for solving linear equations

Step 2: Solve the given equation

Simplification method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Statistics

Decision Maths

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dr}{诲胃} + 迟补苍胃 = 蝉别肠胃$