The solution of the differential equation dy dx = ax+g by+f repre… - Vidyadip

MCQ

The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=\frac{\text{ax}+\text{g}}{\text{by}+\text{f}}$ represents a circle when,

A
$\text{a}=\text{b}$
✓
$\text{a}=-\text{b}$
C
$\text{a}=-2\text{b}$
D
$\text{a}=2\text{b}$

✓

Answer

Correct option: B.

$\text{a}=-\text{b}$

b. $a=-b$
Solution:
We have,
$\frac{d y}{d x}=\frac{a x+g}{b y+f}$
$\Rightarrow(b y+f) d y=(a x+g) d x$
Intergrating both sides, we get
$\Rightarrow \int( by + f ) dy =\int( ax + g ) dx$
$\Rightarrow b \frac{ y ^2}{2}+ fy = a \frac{ x ^2}{2}+ gx + C$
$\Rightarrow b \frac{ y ^2}{2}+ fy - a \frac{ x ^2}{2}-g x = C$
$\Rightarrow b y^2+2 f y-a x^2-2 g x-2 C=0$
The above equation resprasents a circle.
Therefore, the coffrcients of $x^2$ and $y^2$ must be equal.
$- a = b$
$\Rightarrow a =- b$

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