Mathematics - Differential Equation Question with Solution | TestHub

MathematicsDifferential EquationApplication (Mixing, Geometric, temp., trajectory)Medium2 minQB

MathematicsMediumsingle choice

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

Answer:

Solution:

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 $\frac{d y}{d x}$ by $- \frac{d x}{d y}$ ,

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.

Stream:JEESubject:MathematicsTopic:Differential EquationSubtopic:Application (Mixing, Geometric, temp., trajectory)

⏱ 2mℹ️ Source: QB

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