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Changing the Independent Variable in a DE | JEE

JEE Maths question with a full step-by-step solution.

Question

If the independent variable

x

is changed to

y

, then the differential equation

x\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 - \dfrac{dy}{dx} = 0

is transformed to

x\dfrac{d^2x}{dy^2} + \left(\dfrac{dx}{dy}\right)^2 = k

, where

k

is a number. Then

k

equals:

0

1

correct

-1

2

Solution

Use the change-of-variable formulas

\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}, \qquad \frac{d^2y}{dx^2} = -\frac{\frac{d^2x}{dy^2}}{\left(\frac{dx}{dy}\right)^3}.

Substitute into

x\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 - \dfrac{dy}{dx} = 0

(write

x' = \frac{dx}{dy}

x'' = \frac{d^2x}{dy^2}

-\frac{x\,x''}{(x')^3} + \frac{1}{(x')^3} - \frac{1}{x'} = 0.

Multiply by

-(x')^3

x\,x'' - 1 + (x')^2 = 0 \Rightarrow x\frac{d^2x}{dy^2} + \left(\frac{dx}{dy}\right)^2 = 1.

\therefore k = 1

. Correct answer: (2)

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