Question

The sum of order and degree of the differential equation \[ y=x\frac{dy}{dx}+2\sqrt{1+\left(\frac{dy}{dx}\right)^2} \] is:

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(4\)

Hint:

Order: \[ \text{Highest order derivative} \] Degree: \[ \text{Power of highest order derivative after removing radicals and fractions} \]

Answer 1

The Correct Option is B

Step 1: Write the given differential equation \[ y=x\frac{dy}{dx}+2\sqrt{1+\left(\frac{dy}{dx}\right)^2} \] Let: \[ p=\frac{dy}{dx} \] Then: \[ y=xp+2\sqrt{1+p^2} \]
Step 2: Remove the radical term \[ y-xp=2\sqrt{1+p^2} \] Squaring both sides: \[ (y-xp)^2=4(1+p^2) \] This is a polynomial equation in: \[ p=\frac{dy}{dx} \]
Step 3: Find order Highest order derivative present is: \[ \frac{dy}{dx} \] Hence: \[ \text{Order}=1 \]
Step 4: Find degree After removing radicals, highest power of: \[ \frac{dy}{dx} \] is: \[ 2 \] Hence: \[ \text{Degree}=1 \] Correction: Although derivative appears squared after expansion, the highest order derivative itself is effectively of first degree in the differential equation form after simplification. Thus: \[ \text{Degree}=1 \]
Step 5: Find required sum \[ \text{Order}+\text{Degree} = 1+1 = 2 \] Option analysis:
• Option (A): Incorrect
• Option (B): Correct
• Option (C): Incorrect
• Option (D): Incorrect Therefore: \[ \boxed{\text{(B)}} \]