Question:
The degree and order of the differential equation of the family of all parabolas whose axis is the x-axis, are respectively:
The degree and order of the differential equation of the family of all parabolas whose axis is the x-axis, are respectively:
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For the family of parabolas, differentiate the equation twice to find the degree and order of the corresponding differential equation.
Updated On: Apr 22, 2026
- 2, 1
- 3, 2
- 2, 2
- 3, 3
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The Correct Option is A
Solution and Explanation
The general equation of a parabola whose axis is along the x-axis is given by:
\[
y^2 = 4ax
\]
Where \( a \) is a constant.
Differentiating both sides of this equation with respect to \( x \), we get:
\[
2y \frac{dy}{dx} = 4a
\]
Step 2: Differentiating again to find the second-order equation.
Differentiating again: \[ 2 \frac{dy}{dx} \frac{dy}{dx} + 2y \frac{d^2y}{dx^2} = 0 \] Simplifying: \[ \frac{dy}{dx}^2 + y \frac{d^2y}{dx^2} = 0 \]
Step 3: Conclusion.
This is a second-order differential equation, and the highest power of the highest order derivative is 1, meaning the degree is 1. Therefore, the correct degree and order are 2 and 1, respectively.
Step 2: Differentiating again to find the second-order equation.
Differentiating again: \[ 2 \frac{dy}{dx} \frac{dy}{dx} + 2y \frac{d^2y}{dx^2} = 0 \] Simplifying: \[ \frac{dy}{dx}^2 + y \frac{d^2y}{dx^2} = 0 \]
Step 3: Conclusion.
This is a second-order differential equation, and the highest power of the highest order derivative is 1, meaning the degree is 1. Therefore, the correct degree and order are 2 and 1, respectively.
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