Differential Equations Question 309
Question: The solution of the differential equation $ { 1+x\sqrt{( x^{2}+y^{2} )} }dx+{ \sqrt{( x^{2}+y^{2} )}-1 }ydy=0 $ is
Options:
A) $ x^{2}+\frac{y^{2}}{2}+\frac{1}{3}{{( x^{2}+y^{2} )}^{3/2}}=C $
B) $ x-\frac{y^{2}}{3}+\frac{1}{2}{{( x^{2}+y^{2} )}^{1/2}}=C $
C) $ x-\frac{y^{2}}{2}+\frac{1}{3}{{( x^{2}+y^{2} )}^{3/2}}=C $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Rearranging the equation, we have $ dx-ydy+\sqrt{( x^{2}+y^{2} )}(xdx+ydy)=0 $
$ \Rightarrow dx-ydy+\frac{1}{2}\sqrt{( x^{2}+y^{2} )}d(x^{2}+y^{2})=0 $
On integrating, we get $ x-\frac{y^{2}}{2}+\frac{1}{2}\int{\sqrt{t}dt=c,{ t=\sqrt{( x^{2}+y^{2} )} }} $ Or $ x-\frac{y^{2}}{2}+\frac{1}{3}{{( x^{2}+y^{2} )}^{3/2}}=c $
BETA
