Differential Equations Question 304
Question: The general solution of $ y^{2}dx+(x^{2}-xy+y^{2})dy=0 $ is
[EAMCET 2003]
Options:
A) $ {{\tan }^{-1}}( \frac{x}{y} )+\log y+c=0 $
B) $ 2{{\tan }^{-1}}( \frac{x}{y} )+\log x+c=0 $
C) $ \log (y+\sqrt{x^{2}+y^{2}})+\log y+c=0 $
D) $ {{\sinh }^{-1}}( \frac{x}{y} )+\log y+c=0 $
Show Answer
Answer:
Correct Answer: A
Solution:
$ \frac{dx}{dy}+\frac{x^{2}-xy+y^{2}}{y^{2}}=0 $
$ \frac{dx}{dy}+{{( \frac{x}{y} )}^{2}}-( \frac{x}{y} )+1=0 $
Put $ v=x/y $
Therefore $ x=vy $
Therefore $ \frac{dx}{dy}=v+y\frac{dv}{dy} $
$ v+y\frac{dv}{dy}+v^{2}-v+1=0 $
Therefore $ \frac{dv}{v^{2}+1}+\frac{dy}{y}=0 $
Therefore $ \int{\frac{dv}{v^{2}+1}+\int{\frac{dy}{y}=0}} $
Therefore $ {{\tan }^{-1}}(v)+\log y+C=0 $
Therefore $ {{\tan }^{-1}}(x/y)+\log y+c=0 $ .
BETA
