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Question: If \(\frac { d y } { d x } = \frac { x y + y } { x y + x }\), then the solution of the differential ...

If dydx=xy+yxy+x\frac { d y } { d x } = \frac { x y + y } { x y + x }, then the solution of the differential equation is

A

y=xex+cy = x e ^ { x } + c

B

y=ex+cy = e ^ { x } + c

C

y=Axex−yy = A x e ^ { x - y }

D

y=x+Ay = x + A

Answer

y=Axex−yy = A x e ^ { x - y }

Explanation

Solution

dydx=xy+yxy+x\frac { d y } { d x } = \frac { x y + y } { x y + x } ⇒ (1+yy)dy=(1+xx)dx\left( \frac { 1 + y } { y } \right) d y = \left( \frac { 1 + x } { x } \right) d x

On integrating both sides, we get

log⁡y+y=log⁡x+x+log⁡A\log y + y = \log x + x + \log A

⇒ log⁡(yAx)=x−y\log \left( \frac { y } { A x } \right) = x - y⇒y=Axex−yy = A x e ^ { x - y }.