First slide

Applications of differential equations

Question

$Orthogonal trajectories of family of the curve x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}, whose ' a' is an arbitrary constant is$

Moderate

A

$x^{2/3} {-y}^{2/3} =c$
B

$x^{4/3} {-y}^{4/3} =c$
C

$x^{4/3} {+y}^{4/3} =c$
D

$x^{1/3} {-y}^{1/3} =c$

Solution

$\begin{array}{l} \frac{2}{3} x^{\frac{- 1}{3}} + \frac{2}{3} y^{\frac{- 1}{3}} \frac{d y}{d x} = 0 \to \frac{d y}{d x} = - \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}} \\ Replace \frac{d y}{d x} by \frac{- d x}{d y} \\ ∴ \frac{d x}{d y} = \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}} \Rightarrow \int x^{\frac{1}{3}} d x = \int y^{\frac{1}{3}} d y \\ \Rightarrow x^{\frac{4}{3}} - y^{\frac{4}{3}} = c \end{array}$

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