Differential Equations Question 303
Question: Which of the following does not represent the orthogonal trajectory of the system of curves $ {{( \frac{dy}{dx} )}^{2}}=\frac{a}{x} $
Options:
A) $ 9a{{(y+c)}^{2}}=4x^{3} $
B) $ y+c=\frac{-2}{3\sqrt{a}}{x^{3/2}} $
C) $ y+c=\frac{2}{3\sqrt{a}}{x^{3/2}} $
D) All are orthogonal trajectories
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Answer:
Correct Answer: D
Solution:
[d] The family of curves which are orthogonal (i.e. intersect at right angles) to a given system of curves is obtained by substitute $ -\frac{dx}{dy} $ for $ \frac{dy}{dx} $ in the differential equation of the given system, The given differential equation is $ {{( \frac{dy}{dx} )}^{2}}=\frac{a}{x} $
Replacing $ \frac{dy}{dx} $ by $ -\frac{dx}{dy}, $ we get $ {{( \frac{dx}{dy} )}^{2}}=\frac{a}{x}\Rightarrow {{( \frac{dy}{dx} )}^{2}}=\frac{x}{a}\Rightarrow \frac{dy}{dx}=\pm \sqrt{\frac{x}{a}.} $ Integrating we get, $ y+c=\pm \frac{2}{3\sqrt{a}}{x^{3/2}} $ - (i)
$ \Rightarrow {{(y+c)}^{2}}=\frac{4}{9a}x^{3}\Rightarrow 9a{{(y+c)}^{2}}=4x^{3} $
.. (ii) From (i) and (ii) all of the first three given option represent required equations.
BETA
