Differential Equations Question 233
Question: The differential equations of all conies whose axes coincide with the co-ordinate axis
Options:
A) $ xy\frac{d^{2}y}{dx^{2}}+x{{( \frac{dy}{dx} )}^{2}}+y\frac{dy}{dx}=0 $
B) $ xy\frac{d^{2}y}{dx^{2}}+x{{( \frac{dy}{dx} )}^{2}}+x\frac{dy}{dx}=0 $
C) $ xy\frac{d^{2}y}{dx^{2}}+x{{( \frac{dy}{dx} )}^{2}}-y\frac{dy}{dx}=0 $
D) $ xy\frac{d^{2}y}{dx^{2}}-x{{( \frac{dy}{dx} )}^{2}}+y\frac{dy}{dx}=0 $
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Answer:
Correct Answer: C
Solution:
[c] Any conic whose axes coincide with coordinate axis is $ ax^{2}+by^{2}=1 $
(i) Diff. both sides w.r.t. $ ‘x’ $ , we get $ 2ax+2by\frac{dy}{dx}=0 $ i.e., $ ax+by\frac{dy}{dx}=0 $ (ii) Diff. again, $ a+b( y\frac{d^{2}y}{dx^{2}}+{{( \frac{dy}{dx} )}^{2}} )=0 $ (iii) From (ii), $ \frac{a}{b}=-\frac{ydy/dx}{x} $ From (iii), $ \frac{a}{b}=-( y\frac{d^{2}y}{dx^{2}}+{{( \frac{dy}{dx} )}^{2}} ) $
$ \therefore \frac{y\frac{dy}{dx}}{x}=y\frac{d^{2}y}{dx^{2}}+{{( \frac{dy}{dx} )}^{2}} $
$ \Rightarrow xy\frac{d^{2}y}{dx^{2}}+x{{( \frac{dy}{dx} )}^{2}}-y\frac{dy}{dx}=0 $
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