Question

(2x - y) dx = (x-y) dy solve the differential equations ​

Answer 1

SOLUTION

TO SOLVE

The differential equation

$\sf{(2x - y)dx = (x - y)dy}$

EVALUATION

Here the given differential equation is

$\sf{(2x - y)dx = (x - y)dy}$

We solve as below

$\sf{(2x - y)dx = (x - y)dy}$

$\sf{ \implies \: 2x \: dx- y \: dx = x \: dy- y \: dy}$

$\sf{ \implies \: 2x \: dx- y \: dx - x \: dy + y \: dy = 0}$

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

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

On integration we get

$\displaystyle\sf{ \int 2x \: dx- \int d \: (x y ) + \int y \: dy = 0}$

$\displaystyle\sf{ \implies 2. \frac{ {x}^{2} }{2} - (x y ) + \frac{ {y}^{2} }{2} = \frac{c}{2} }$

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

Where C is integration constant

FINAL ANSWER

Hence the required solution is

$\boxed{ \: \: \displaystyle\sf{ 2 {x}^{2} -2 xy + {y}^{2} = c} \: \: }$

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