Differential Equations Question 286
Question: If $ y+x\frac{dy}{dx}=x\frac{\phi (xy)}{\phi ‘(xy)} $ then $ \phi (xy) $ is equation to
Options:
A) $ k{e^{x^{2}/2}} $
B) $ k{e^{y^{2}/2}} $
C) $ k{e^{xy/2}} $
D) $ ke^{xy} $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Put $ xy=v $
$ \therefore y+x\frac{dy}{dx}=\frac{dv}{dx}\Rightarrow \frac{dv}{dx}=x\frac{\phi (v)}{\phi ‘(v)} $
$ \therefore \frac{\phi ‘(v)}{\phi (v)}dv=xdx. $ Integrating, we get $ \log \phi (v)=\frac{x^{2}}{2}+\log k\Rightarrow \log \frac{\phi (v)}{k}=\frac{x^{2}}{2} $ or $ \phi (v)=k{e^{x^{2}/2}}\Rightarrow \phi (xy)=k{e^{x^{2}/2}} $
BETA
