Differential Equations Question 261
Question: If for the differential equation $ y’=\frac{y}{x}+\phi ( \frac{x}{y} ), $ the general solution is $ y=\frac{x}{\log | Cx |}, $ then $ \phi (x/y) $ is given by
Options:
A) $ -x^{2}/y^{2} $
B) $ -y^{2}/x^{2} $
C) $ x^{2}/y^{2} $
D) $ -y^{2}/x^{2} $
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Putting $ v=y/x $ so that $ x\frac{dv}{dx}+v=\frac{dv}{dx} $ We have $ x\frac{dv}{dx}+v=v+\phi (1/v) $
$ \Rightarrow \frac{dv}{\phi (1/v)}=\frac{dx}{x};\Rightarrow \log | Cx |=\int{\frac{dv}{^{\phi (1/v)}}} $ But $ y=\frac{x}{\log | Cx |} $ is the general solution, So $ \frac{x}{y}=\frac{1}{v}=\log | Cx |=\int{\frac{dv}{\phi (1/v)}} $
$ \Rightarrow \phi (1/v)=-1/v^{2} $ (Differentiating w.r.t.v both sides)
$ \Rightarrow \phi (x/y)=-y^{2}/x^{2} $
BETA
