Question

If $m$ is order and $n$ is degree of the differential equation $y = x\frac{dy}{dx} + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$, then the value of $m + n$ is

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

Whenever you see a differential equation with a derivative inside a square root or fraction, your very first step must always be algebraically isolating and eliminating that radical by squaring. You cannot determine the true degree otherwise.

Answer 1

The Correct Option is B

Step 1: Recall the two words.
Order is the highest derivative present. Degree is the power of that highest derivative, but only after the equation is free of roots on the derivatives.

Step 2: Write the equation.
\[ y = x\frac{dy}{dx} + \sqrt{a^2\left(\frac{dy}{dx}\right)^2 - b^2} \]
Step 3: Move the root alone.
\[ y - x\frac{dy}{dx} = \sqrt{a^2\left(\frac{dy}{dx}\right)^2 - b^2} \]
Step 4: Square both sides.
\[ \left(y - x\frac{dy}{dx}\right)^2 = a^2\left(\frac{dy}{dx}\right)^2 - b^2 \] Now there is no root, so we can read off the degree.

Step 5: Read order and degree.
The highest derivative is $\frac{dy}{dx}$, a first derivative, so order $m = 1$. The highest power of it is 2, so degree $n = 2$.

Step 6: Add them.
\[ m + n = 1 + 2 = 3 \] Always remove roots on the derivatives before counting the degree. \[ \boxed{m + n = 3 \text{ (Option 2)}} \]