Differential Equations Question 64
Question: The solution of differential equation $ y-x\frac{dy}{dx}=a( y^{2}+\frac{dy}{dx} ) $ is
[MP PET 2002]
Options:
A) $ (x+a)(x+ay)=cy $
B) $ (x+a)(1-ay)=cy $
C) $ (x+a)(1-ay)=c $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ y-x\frac{dy}{dx}=a( y^{2}+\frac{dy}{dx} ) $
Therefore $ y-ay^{2}=a\frac{dy}{dx}+x\frac{dy}{dx} $
Therefore $ y(1-ay)=( a+x ).\frac{dy}{dx} $
Therefore $ \frac{dx}{(a+x)}=\frac{dy}{y(1-ay)} $
Integrating both sides, $ \int{\frac{dx}{(a+x)}=}\int{\frac{dy}{y(1-ay)}} $
Therefore $ \int{\frac{dx}{a+x}=\int{[ \frac{1}{y}+\frac{a}{(1-ay)} ]dx}} $
$ \log (a+x)=\log y+\frac{a\log (1-ay)}{-a} $
Therefore $ \log (a+x)=\log y-\log (1-ay)+\log c $
Therefore $ \log (x+a)(1-ay)=\log cy $
Therefore $ (x+a)(1-ay)=cy $ .
BETA
