Order and Degree of a Differential Equation

A differential equation is an equation involving an independent variable, a dependent variable, and one or more derivatives of the dependent variable with respect to the independent variable.

In this topic, the main goal is to understand two basic ideas: the order of a differential equation and the degree of a differential equation.

The order of the highest differential coefficient (or the highest order derivative appearing in a differential equation) is the order of the differential equation.

The highest exponent of the highest derivative is called the degree of a differential equation, provided exponents of each derivative and an unknown variable appearing in the differential equation are non-negative integers.

Find the order and degree, if defined, of each of the following differential equations:

1.  \[\frac{dy}{dx} - \cos x = 0\]
2. \[xy \frac{d^{2}y}{dx^{2}} + x \left( \frac{dy}{dx} \right)^{2} - y \frac{dy}{dx} = 0\]
3. \[y''' + y^{2} + e^{y'} = 0\]

Solution:

1. The highest order derivative in the given differential equation is \[\frac{dy}{dx}\], so its order is one.

   Order = 1, Degree = 1

2. The highest order derivative in the given differential equation is \[\frac{d^{2}y}{dx^{2}}\], so its order is two.
   

   Order = 2, Degree = 1

3. The highest order derivative  in the given differential equation is \[y'''\], so its order is three.
   

   Order = 3, Degree = 1

• Order = highest derivative order.

• Degree = power of highest derivative.

• Degree exists only for polynomial equations in derivatives.

• Always check polynomial condition before stating the degree.