The order of differential equation of family of circles in a plane is $m$ and highest power of second differential $\left(\frac{d^2y}{dx^2}\right)$ is $n$ then $(m+n)$ ____.
Step-by-Step Solution
Key Concept: The order of a differential equation equals the number of arbitrary constants in the family equation, which is 3 for circles, and eliminating constants through differentiation creates terms with $\left(\frac{d^2y}{dx^2}\right)^2$.
The general equation of a family of circles in a plane is $(x-h)^2 + (y-k)^2 = r^2$, which contains three arbitrary constants: $h$, $k$, and $r$. To eliminate these three constants, we need to differentiate three times, giving us a third-order differential equation, so $m = 3$. Differentiating successively and eliminating constants yields a differential equation where the highest power of $\frac{d^2y}{dx^2}$ is 2, so $n = 2$. Therefore, $m + n = 3 + 2 = 5$.
Correct Answer: 5