A) \[2({{x}^{2}}-{{y}^{2}})=3x\]
B) \[2({{x}^{2}}-{{y}^{2}})=6y\]
C) \[x({{x}^{2}}-{{y}^{2}})=6\]
D) \[x({{x}^{2}}+{{y}^{2}})=10\]
Correct Answer: A
Solution :
\[\frac{dy}{dx}=\frac{{{x}^{2}}+{{y}^{2}}}{2xy}\]. Put \[y=vx\] Þ \[v+x.\frac{dv}{dx}=\frac{{{x}^{2}}+{{v}^{2}}{{x}^{2}}}{2v{{x}^{2}}}\] \[\frac{2v}{1-{{v}^{2}}}.dv=\frac{dx}{x}\] Integrating both sides, \[-\log (1-{{v}^{2}})=\log x+\log c\] \[-\log \left( 1-\frac{{{y}^{2}}}{{{x}^{2}}} \right)=\log x+\log c\] ?..(i) This passes through \[(2,\,1)\] \[-\log \,\left( 1-\frac{1}{4} \right)=\log 2+\log c\] Þ \[c=\frac{2}{3}\] From equation (i), \[\log \left( \frac{{{x}^{2}}}{{{x}^{2}}-{{y}^{2}}} \right)=\log xc\] \[2\,({{x}^{2}}-{{y}^{2}})=3x\].
You need to login to perform this action.
You will be redirected in
3 sec
