Differential Equations Question 268
Question: The curve satisfying the equation $ \frac{dy}{dx}=\frac{y(x+y^{3})}{x(y^{3}-x)} $ and passing through the point (4, -2) is
Options:
A) $ y^{2}=-2x $
B) $ y=-2x $
C) $ y^{3}=-2x $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ (xy^{3}-x^{2})dy-(xy+y^{4})dx=0 $
$ \Rightarrow y^{3}(xdy-ydx)-x(xdy+ydx)=0 $
$ \Rightarrow x^{2}y^{3}\frac{(xdy-ydx)}{x^{2}}-x(xdy+ydx)=0 $
$ \Rightarrow x^{2}y^{3}d( \frac{y}{x} )-xd(xy)=0 $ Dividing by $ x^{3}y^{2}, $ we get $ \frac{y}{x}d( \frac{y}{x} )-\frac{d(xy)}{x^{2}y^{2}}=0 $ Now, integrating $ \frac{1}{2}{{( \frac{y}{x} )}^{2}}+\frac{1}{xy}=c $ It passes through the point $ (4,-2). $
$ \Rightarrow \frac{1}{8}-\frac{1}{8}=c\Rightarrow c=0\therefore y^{3}=-2x $
BETA
