Question

14. Solve the differential equation
(4x+3y +1)dx +(3x +2y+1)dy = 0 and show that it represents a family of
hyperbalas having as asymptotes the lines x+y=0,2x+y+1=0.​

Answer 1

SOLUTION

TO DETERMINE

To Solve the differential equation

(4x+3y +1)dx +(3x +2y+1)dy = 0

and to show that it represents a family of hyperbolas having as asymptotes the lines x+y=0 , 2x+y+1=0.

EVALUATION

Here the given differential equation is

(4x+3y +1)dx +(3x +2y+1)dy = 0

$\sf{ \implies \: 4x \: dx + 3(y \: dx + x \: dy) + 2y \: dy + dx + dy = 0}$

$\sf{ \implies \: 4x \: dx + 3d(xy ) + 2y \: dy + dx + dy = 0}$

On integration we get

$\displaystyle\sf{ \implies \int 4x \: dx + 3 \int d(xy ) + 2 \int y \: dy + \int \: dx + \int \: dy = 0}$

$\displaystyle\sf{ \implies 2 {x}^{2} + 3 xy + {y}^{2} + x + y= c}$

Where C is integration constant

The above equation represents the required solution

On simplification we get

$\displaystyle\sf{ \implies (x + y)(2x + y + 1)= c}$

From which it is clear that it represents a family of hyperbolas having as asymptotes the lines x+y=0 , 2x+y+1=0.

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