The orthogonal trajectories of the family of curves a n − 1 y = x n are given by Options: A . &nbsp x n + n 2 y =constant B . &nbsp n y 2 + x 2 =constant C . &nbsp n 2 x + y n =constant D . &nbsp n 2 x − y n =constant

Answer: Option B : B Differentiating, we have a n − 1 d y d x = n x n − 1 ⇒ a n − 1 = n x n − 1 d x d y Putting this value in the given equation, we have n x n − 1 d x d y y = x n Replacing d y d x by − d y d y , we have n y = − x d x d y ⇒ n y d y + x d x = 0 ⇒ n y 2 + x 2 =constant. Which is the required family of orthogonal trajectories.

The orthogonal trajectories of the family of curves an−1y=xn are g

Discussion Forum : Differential Equations

Question -

The orthogonal trajectories of the family of curves $a^{n - 1} y = x^{n}$ are given by

Options:

A .

x^{n} + n^{2} y

=constant

B .

n y^{2} + x^{2}

=constant

C .

n^{2} x + y^{n}

=constant

D .

n^{2} x - y^{n}

=constant

Answer: Option B
:
B
Differentiating, we have

a^{n - 1} \frac{d y}{d x} = n x^{n - 1} \Rightarrow a^{n - 1} = n x^{n - 1} \frac{d x}{d y}

Putting this value in the given equation, we have

n x^{n - 1} \frac{d x}{d y} y = x^{n}

Replacing

\frac{d y}{d x}

- \frac{d y}{d y}

, we have

n y = - x \frac{d x}{d y}

\Rightarrow n y d y + x d x = 0 \Rightarrow n y^{2} + x^{2}

=constant. Which is the required family of orthogonal trajectories.

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