Theory

Ordinary Differential Equations (ODEs) describe the relationship between a function and its derivatives. They play a fundamental role in modeling real-world systems involving motion, heat flow, population growth, electrical circuits, and many other scientific and engineering applications.

In many cases, ODEs do not have closed-form analytical solutions. To handle such equations, approximation methods are used. One powerful approach is the Power Series Method, where the unknown solution is expressed as a polynomial in terms of the independent variable.

A power series representation of a function around a point ( x_0 ) is given by:

y(x) = a₀ + a₁(x − x₀) + a₂(x − x₀)² + a₃(x − x₀)³ + ⋯

This form allows us to convert the differential equation into a sequence of algebraic equations by substituting the series into the ODE and matching coefficients of like powers. The coefficients ( a₀, a₁, a₂, ... ) are then determined using:

• The differential equation itself
• Given initial conditions

A special case of the power series method is the Taylor Series Method, where the solution is expanded using successive derivatives evaluated at a specific point:

y(x) ≈ y(x₀) + y′(x₀)(x − x₀) + y″(x₀)/2! (x − x₀)² + ⋯

This technique provides:

• A step-by-step numerical approximation
• Smooth and highly accurate representation near the expansion point

Power series solutions are particularly useful when:

• The function is smooth and differentiable
• Solutions around ordinary points of the differential equation are needed
• Analytical solutions are difficult or impossible to derive

Applications of power series ODE solutions include:

• Quantum mechanics and wave equations
• Fluid dynamics
• Electrical and mechanical vibration analysis
• Numerical simulation and scientific computing

Thus, the power series method offers a flexible and robust technique to approximate solutions to many nonlinear or complex differential equations encountered in practical systems.