Differential Equations
An equation involving one dependent variable, one or more independent variable and the differential coefficient (derivatives) of dependent variable with respect to independent variable is called differential equations.
Examples:
1. \(\frac{dy}{dx}=3x+2y\),
2. \(\frac{{{d}^{2}}y}{d{{x}^{2}}}+9=0\).
Types of differential equations
1. Ordinanry differential equation: A differential equations is said to be an ordinary differential equation. If it contains only one independent variable and ordinary differentiation with respect to the independent variable,
Example: \(\frac{dy}{dx}=3x+2\),\(\frac{{{d}^{2}}y}{d{{x}^{2}}}+9y=0\).
2. Partial differential equation: A differential equation is said to be a partial differential equation. If it contains at least two independent variable and partial differentiation with respect to either of these independent variable.
Example: \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=0\).
Order of a differential equation: The order of the height derivative involved in an ordinary differential equation is called the order of the differential equation.
Example: find the order of differential equation \(x{{\left( \frac{dy}{dx} \right)}^{2}}+8y=0\).
Solution: Given \(x{{\left( \frac{dy}{dx} \right)}^{2}}+8y=0\).
Given equation highest differentiation is order,
Therefore order of differential equation \(x{{\left( \frac{dy}{dx} \right)}^{2}}+8y=0\)is 1.
Degree of a differential equation: The degree of the height derivative involved in a ordinary differential equation, when the equation has been expressed in the form of polynomial in the height derivative by eliminating radicals and fraction power of the derivative is called degree of the differential equation.
Example: Find the degree of differential equation \(\frac{{{d}^{2}}y}{d{{x}^{2}}}+3{{\left( \frac{dy}{dx} \right)}^{1}}=0\).
Solution: \(\frac{{{d}^{2}}y}{d{{x}^{2}}}+3{{\left( \frac{dy}{dx} \right)}^{1}}=0\).
Given differential equation height derivative power is degree the degree of differential equation \(\frac{{{d}^{2}}y}{d{{x}^{2}}}+3{{\left( \frac{dy}{dx} \right)}^{1}}=0\) is 1.