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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Proved

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 $a_{2} = k a_{1} ...... (1)$ ,

and $b_{2} = k b_{1} ...... (2)$

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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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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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

Pure Maths

Decision Maths

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Calculus

Applied 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 $b_{2} = k b_{1}$ .]