Differential Equations Question 276
Question: The solution of the differential equation $ \frac{dy}{dx}=\frac{y}{x}+\frac{\varphi ( \frac{y}{x} )}{{\varphi }’( \frac{y}{x} )} $ is
[DCE 2002]
Options:
A) $ \varphi ( \frac{y}{x} )=kx $
B) $ x\varphi ( \frac{y}{x} )=k $
C) $ \varphi ( \frac{y}{x} )=ky $
D) $ y\varphi ( \frac{y}{x} )=k $
Show Answer
Answer:
Correct Answer: A
Solution:
$ \frac{dy}{dx}=\frac{y}{x}+\frac{\varphi ( \frac{y}{x} )}{{\varphi }’( \frac{y}{x} )} $ . Put $ y=vx $
Therefore $ \frac{dy}{dx}=v+x\frac{dv}{dx} $
\ The given differential equation becomes
$ v+x\frac{dv}{dx}=v+\frac{\varphi (v)}{{\varphi }’(v)} $
Therefore $ \frac{{\varphi }’(v)}{\varphi (v)}dv=\frac{dx}{x} $
Therefore $ \log \varphi (v)=\log x+\log k $
Therefore $ \varphi (v)=kx $
Therefore $ \varphi ( \frac{y}{x} )=kx $ .
BETA
