# Application of Differential Equation

Application of Differential Equation

Differential equation can be defined as a mathematical equation for an unknown function of one or more variables and the Variables are related to the value of the function itself and its derivation of various order. Differential equation plays an important role in the field of math, physics, engineering etc.

Some useful relations:

1. Slope of tangent = $\dfrac{dy}{dx}$
2. Slope of normal = $- \dfrac{dx}{dy}$
3. Length of tangent = $\dfrac{y}{\left(\frac{dy}{dx} \right)^2} \sqrt{1 + \left(\dfrac{dy}{dx} \right)^2}$
4. Length of normal = $y \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2}$
5. Equation of tangent at $p (x_1, y_1)$ is $Y- y_1 = \dfrac{dy}{dx} (X- x_1)$
6. Equation of Normal $Y- y_1 = \dfrac{dx}{dy} (X- x_1)$
7. Radius of curvature $\rho = \dfrac{\{1 + \left(\dfrac{dy}{dx} \right)^2 \}^\frac{3}{2}}{\dfrac{d^2y}{dx^2}}$ or $\rho = \dfrac{ds}{d\Psi}$

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