Differential Equations Question 11
Question: The solution of the differential equation $ \frac{dy}{dx}=\frac{x-y+3}{2(x-y)+5} $ is
Options:
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
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ x-y=v $ and $ \frac{dy}{dx}=1-\frac{dv}{dx}, $ thus the equation reduces to $ \frac{dv}{dx}=\frac{v+2}{2v+5} $
Therefore $ \int _{{}}^{{}}{\frac{2v+5}{v+2}}dv=\int _{{}}^{{}}{dx} $
Therefore $ \int _{{}}^{{}}{[ 2+\frac{1}{(v+2)} ]}dv=\int _{{}}^{{}}{dx} $
Therefore $ 2v+\log (v+2)=x+c $
or $ 2(x-y)+\log (x-y+2)=x+c $ .
BETA
