Differential Equations Question 326
Question: What is the solution of the differential equation $ (x+y)(dx-dy)=dx+dy $ -
Options:
A) $ x+y+ln(x+y)=c $
B) $ x-y+ln(x+y)=c $
C) $ y-x+ln(x+y)=c $
D) $ y-x-ln(x-y)=c $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Differential equation is $ (x+y)(dx-dy)=dx+dy $
dividing by dx on both the sides
$ (x+y)( 1-\frac{dy}{dx} )=1+\frac{dy}{dx} $
Putting $ x+y=v $
$ 1+\frac{dy}{dx}=\frac{dv}{dx} $ and $ \frac{dy}{dx}=\frac{dv}{dx}-1 $
The equation changes to $ v{ 1-( \frac{dv}{dx}-1 ) }=\frac{dv}{dx};v( 2-\frac{dv}{dx} )=\frac{dv}{dx} $
$ 2v-v\frac{dv}{dx}=\frac{dv}{dx};2v=(1+v)\frac{dv}{dx} $
$ ( \frac{1+v}{v} )dv=2dx $ or, $ ( \frac{1}{v}+1 )dv=2dx $
Integrating on both the sides.
$ \int{\frac{dv}{v}+\int{dv=2\int{dx+c}}} $
$ \log v+v=2x+c $
Putting $ v=x+y $
$ \log (x+y)+x+y=2x+c $
or, $ \log (x+y)+y-x=c $
or, $ y-x+\log (x+y)=c $
BETA
