# Order and Degree of Differential Equations

### Order and Degree of Differential Equations:

1. Order of a Differential Equation:

• The order of a differential equation is the highest derivative of the dependent variable present in the equation.
• It indicates the level of complexity in the equation.
• Examples:
• $\frac{{d}^{2}y}{d{x}^{2}}-3\frac{dy}{dx}+2y=0$ is a second-order differential equation.
• $\frac{dy}{dx}+y={e}^{x}$ is a first-order differential equation.
2. Degree of a Differential Equation:

• The degree of a differential equation is the power to which the highest-order derivative is raised after the equation is made free from radicals and fractions.
• For a polynomial equation in derivatives, the highest power of the derivative is the degree.
• Example:
• $\frac{{d}^{3}y}{d{x}^{3}}-2{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{2}+4\frac{dy}{dx}=0$ has the highest power of the derivative as 3, making it a third-degree differential equation.
3. Importance of Order and Degree:

• Help categorize and identify the complexity of differential equations.
• Determine the type of methods and approaches used to solve them.
• Indicate the number of arbitrary constants or functions present in the general solution.
4. Distinguishing Order and Degree:

• Order: Refers to the highest derivative present.
• Degree: Involves the power to which the highest-order derivative is raised in the equation when it's made free from radicals and fractions.
5. Applications:

• The order and degree help in determining the methods used to solve differential equations.
• They aid in understanding the behavior and characteristics of systems modeled by these equations.
6. Solving Based on Order and Degree:

• Equations of higher orders or degrees might require more sophisticated techniques or transformations to solve.
• Lower-order and lower-degree equations often have more straightforward solution methods.

### Example: Order and Degree of a Differential Equation

Consider the differential equation $\frac{{d}^{2}y}{d{x}^{2}}+3\frac{dy}{dx}-2y=0$.

1. Identifying the Order and Degree:

• Order: The highest derivative present in the equation is the second derivative $\frac{{d}^{2}y}{d{x}^{2}}$, so the order of this differential equation is 2.
• Degree: After simplification to eliminate fractions or radicals, determine the highest power of the derivative. In this case, the second derivative is not raised to any power, so the degree of the equation is 1 (since $\frac{{d}^{2}y}{d{x}^{2}}$ is to the power of 1).
2. Result:

• The given differential equation $\frac{{d}^{2}y}{d{x}^{2}}+3\frac{dy}{dx}-2y=0$ is a second-order and first-degree differential equation.