Question

The solution of the differential equation

$\frac { d y } { d x } = \frac { x - y + 3 } { 2 ( x - y ) + 5 }$ is

• A. $2 ( x - y ) + \log ( x - y ) = x + c$
• B. $2 ( x - y ) - \log ( x - y + 2 ) = x + c$
• C. $2 ( x - y ) + \log ( x - y + 2 ) = x + c$
• D. None of these

Answer 1

Answer: (C)

$2 ( x - y ) + \log ( x - y + 2 ) = x + c$

Answer 2

Let $x - y = v$and $\frac { d y } { d x } = 1 - \frac { d v } { d x }$ thus the equation reduces to $\frac { d v } { d x } = \frac { v + 2 } { 2 v + 5 }$⇒$\int \frac { 2 v + 5 } { v + 2 } d v = \int d x$

⇒ $\int \left[ 2 + \frac { 1 } { ( v + 2 ) } \right] d v = \int d x$ ⇒ $2 v + \log ( v + 2 ) = x + c$

or $2 ( x - y ) + \log ( x - y + 2 ) = x + c$ .