Q 2.6-44E Question: Show that equation (13... [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: Q 2.6-44E (page 77)

Question: Show that equation (13) reduces to an equation of the form $\frac{d y}{d x} = G (a x + b y),$
when [Hint: If $a_{1} b_{2} = a_{2} b_{1}$ , then $a_{2} / a_{1} = b_{2} / b_{1} = k,$ so that $a_{2} = k a_{1}$ and $b_{2} = k b_{1}$ .]

Short Answer

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Step by step solution

Analyzing the given statement

Given equation (13) is,

$(a_{1} x + b_{1} y + c_{1}) d x + (a_{2} x + b_{2} y + c_{2}) d y = 0$

Show that the equation (13) reduces to,

$\frac{d y}{d x} = G (a x + b y), when a_{1} b_{2} = a_{2} b_{1} .$

Initial step of the solution

Given that $a_{1} b_{2} = a_{2} b_{1}$ ,

Therefore $a_{2} / a_{1} = b_{2} / b_{1} = k,$ ,

So that $a_{2} = k a_{1} 鈥� ...... (1)$ ,

and $b_{2} = k b_{1} 鈥夆赌夆赌夆赌夆赌夆赌夆赌� ...... (2)$

The values from equations (1) and (2) in equation (13) in the next step.

Using the values from equations (1) and (2) in equation (13) and simplifying it

Substituting values from equations (1) and (2) in equation (13)

$\begin{matrix} (a_{1} x + b_{1} y + c_{1}) d x + (k a_{1} x + k b_{1} y + c_{2}) d y = 0 \\ a_{1} x (d x + k d y) + b_{1} y (d x + k d y) + c_{1} d x + c_{2} d y = 0 \\ (a_{1} x + b_{1} y) (d x + k d y) + c_{1} d x + c_{2} d y = 0 \\ (a_{1} x + b_{1} y) (1 + k \frac{d y}{d x}) + c_{1} + c_{2} \frac{d y}{d x} = 0 \end{matrix}$

$\begin{matrix} a_{1} x + b_{1} y + (a_{1} x + b_{1} y) k \frac{d y}{d x} + c_{1} + c_{2} \frac{d y}{d x} = 0 \\ ((a_{1} x + b_{1} y) k + c_{2}) \frac{d y}{d x} = - (a_{1} x + b_{1} y + c_{1}) \\ \frac{d y}{d x} = \frac{- (a_{1} x + b_{1} y + c_{1})}{(a_{1} x + b_{1} y) k + c_{2}} \end{matrix}$

Hence, it could be reduced to $\frac{dy}{dx} = G (ax + by)$ .

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91影视

Question: Show that equation (13) reduces to an equation of the form $\frac{d y}{d x} = G (a x + b y),$
when [Hint: If $a_{1} b_{2} = a_{2} b_{1}$ , then $a_{2} / a_{1} = b_{2} / b_{1} = k,$ so that $a_{2} = k a_{1}$ and $b_{2} = k b_{1}$ .]

Short Answer

Step by step solution

Analyzing the given statement

Initial step of the solution

Using the values from equations (1) and (2) in equation (13) and simplifying it

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91影视

Question: Show that equation (13) reduces to an equation of the form dydx=G(ax+by), when [Hint: If a1b2=a2b1, then a2/a1=b2/b1=k,so thata2=ka1 and b2=kb1.]

Short Answer

Step by step solution

Analyzing the given statement

Initial step of the solution

Using the values from equations (1) and (2) in equation (13) and simplifying it

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Question: Show that equation (13) reduces to an equation of the form $\frac{d y}{d x} = G (a x + b y),$
when [Hint: If $a_{1} b_{2} = a_{2} b_{1}$ , then $a_{2} / a_{1} = b_{2} / b_{1} = k,$ so that $a_{2} = k a_{1}$ and $b_{2} = k b_{1}$ .]