Introduction to ODE Order and Degree

Introduction to Order and Degree of Ordinary Differential Equations

• An ordinary differential equation (ODE) is an equation containing an unknown function of a single independent variable and its derivatives.
• The order of an ODE is the order of the highest derivative present. A first-order ODE involves only the first derivative, a second-order ODE involves the second derivative, and so on.

Defining Order

• The order of a differential equation is determined by the highest-order derivative present.
• Example: dy/dx + 2y = x, is a first-order ODE.
• Example: d²y/dx² + 5(dy/dx) + 6y = sin(x) is a second-order ODE.

Defining Degree

• The degree of a differential equation corresponds to the power of the highest-order derivative after it has been made free of any radicals and fractions.
• Often, differential equations are presented in a form involving radicals or fractional powers. These are typically simplified. If there are no fractions or radicals, the coefficient of the highest-order derivative is the degree.
• Example: (d²y/dx²)² + 4 (dy/dx)³ + 5y = x is a second-order equation and is of degree 2, since the highest-order derivative, (d²y/dx²), is squared.
• Example: √(d²y/dx²) + 2 (dy/dx) = e^x is of degree 2. (Squaring both sides gives (d²y/dx²) + 2(dy/dx)=e^x)^2, which is of degree 2).
• Example: (d²y/dx²) + 3(dy/dx) + 2y = x², This is a second-order equation of degree 1. No fractions or radicals.

Identifying Order and Degree - Examples

• Consider the following differential equations:
  • (d³y/dx³) + 2(d²y/dx²) + y = 0. Order: 3; Degree: 1
  • (d²y/dx²)³ + 3(dy/dx)² + 5y = x. Order: 2; Degree: 3
  • √(d²y/dx²) + 7 (dy/dx) + 8y = sin x. Order: 2; Degree: 2 (Squaring both sides eliminates the square root).
  • (dy/dx) = 2x + 4. Order: 1, Degree: 1

Important Considerations

• Eliminate any radicals or fractional powers of derivatives before finding degree. This requires raising the entire equation to the appropriate power to clear any fractional exponents.
• Sometimes, algebraic manipulation is needed to identify order and degree when equations aren't presented in a standard form.

Conclusion

• The order of a differential equation dictates the number of initial conditions required for a unique solution.

This quiz covers the fundamentals of ordinary differential equations (ODEs), focusing on their order and degree. Learn to identify the order of an ODE based on its highest derivative and understand how to determine its degree after simplification. Perfect for students studying differential equations.