Question

If order and degree of the differential equation corresponding to the family of curves y² = 4a(x+a)(a is parameter) are m and n respectively, then m+n² =

• A. 3
• B. 4
• C. 5
• D. 2

Answer 1

The Correct Option is C

To solve the problem, we need to find the order $m$ and degree $n$ of the differential equation for the family of curves $y^2 = 4a(x + a)$, where $a$ is a parameter, and compute $m + n^2$.

1. Differentiate the Equation:
Given $y^2 = 4a(x + a)$, differentiate with respect to $x$:
$2y \frac{dy}{dx} = 4a$.
Solve for $a$:
$a = \frac{y}{2} \frac{dy}{dx}$.

2. Eliminate the Parameter:
Substitute $a = \frac{y}{2} \frac{dy}{dx}$ into $y^2 = 4a(x + a)$:
$y^2 = 4 \left( \frac{y}{2} \frac{dy}{dx} \right) \left( x + \frac{y}{2} \frac{dy}{dx} \right)$.
Simplify:
$y^2 = 2y \frac{dy}{dx} \left( x + \frac{y}{2} \frac{dy}{dx} \right)$.
Divide by $y$ (assuming $y \neq 0$):
$y = 2 \frac{dy}{dx} \left( x + \frac{y}{2} \frac{dy}{dx} \right)$.
Expand:
$y = 2x \frac{dy}{dx} + y \left( \frac{dy}{dx} \right)^2$.

3. Determine Order and Degree:
The differential equation is $y = 2x \frac{dy}{dx} + y \left( \frac{dy}{dx} \right)^2$.
The highest-order derivative is $\frac{dy}{dx}$, so the order is $m = 1$.
The highest power of $\frac{dy}{dx}$ is $\left( \frac{dy}{dx} \right)^2$, so the degree is $n = 2$.

4. Compute $m + n^2$:
Calculate:
$m + n^2 = 1 + 2^2 = 1 + 4 = 5$.

Final Answer:
The value of $m + n^2$ is $5$.