Question

The degree of differential equation of all curves having normal of constant length \(c\) is

• A. 1
• B. 3
• C. 4
• D. None of these

Hint:

If normal length is constant, differential equation becomes \(y^2(1+(y')^2)=c^2\), so degree is 2. If 2 not given, choose None of these.

Answer 1

The Correct Option is D

Step 1: Use normal length formula.
For curve \(y=f(x)\), length of normal segment is:
\[ N = y\sqrt{1+\left(\frac{dy}{dx}\right)^2} \] Given normal is constant:
\[ y\sqrt{1+\left(\frac{dy}{dx}\right)^2}=c \] Step 2: Remove square root.
\[ y^2\left(1+\left(\frac{dy}{dx}\right)^2\right)=c^2 \] This is a first order differential equation.
Step 3: Degree definition.
Degree is the power of highest derivative after removing radicals/fractions.
Here highest derivative is \(\frac{dy}{dx}\) and it appears as power 2.
So degree is 2.
Step 4: Match options.
2 is not in options, hence None of these.
Final Answer: \[ \boxed{\text{None of these}} \]