Differential Equations Question 201
Question: The solution of differential equation $ yy’=x( \frac{y^{2}}{x^{2}}+\frac{f(y^{2}/x^{2})}{f’(y^{2}/x^{2})} ) $ is
Options:
A) $ f(y^{2}/x^{2})=cx^{2} $
B) $ x^{2}f(y^{2}/x^{2})=c^{2}y^{2} $
C) $ x^{2}f(y^{2}/x^{2})=c $
D) $ f(y^{2}/x^{2})=cy/x $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] The given equation can be written as $ \frac{y}{x}\frac{dy}{dx}={ \frac{y^{2}}{x^{2}}+\frac{f(y^{2}/x^{2})}{f’(y^{2}/x^{2})} } $ The above equation is a homogeneous equation. Putting $ y=vx, $ we get $ v[ v+x\frac{dv}{dx} ]=v^{2}+\frac{f(v^{2})}{f’(v^{2})} $ or $ vx\frac{dv}{dx}=\frac{f(v^{2})}{f’(v^{2})} $
(variable separable) or $ \frac{2vf’(v^{2})}{f(v^{2})}dv=2\frac{dx}{x} $ Now, integrating both sides, we get $ \log f(v^{2})=logx^{2}+\log c[logc=constant] $ or $ \log f(v^{2})=logcx^{2}orf(v^{2})=cx^{2} $ or $ f(y^{2}/x^{2})=cx^{2} $
BETA
