Solving Algebraic Differential Equations with MATLAB

I developed a MATLAB program to solve systems of algebraic differential equations. The implementation begins with analyzing the equation system to determine initial conditions for each unknown variable and defining the time domain for numerical integration. The core solving process employs several numerical methods including the Euler method for basic implementation and Runge-Kutta methods (such as ode45) for higher precision. However, during result analysis and validation, I identified potential accuracy limitations in numerical approaches for certain stiff equations. This led to incorporating symbolic computation tools like Maple's dsolve function through MATLAB's symbolic math toolbox. The symbolic approach enables exact solutions where possible and provides deeper mathematical insight into equation behavior. Key implementation aspects include: - Using MATLAB's ode solvers with appropriate tolerances and step sizes - Implementing custom Euler method with fixed/variable step control - Applying symbolic differentiation and equation simplification - Validating solutions through residual analysis and cross-method comparison - Visualizing results using MATLAB's plotting functions for solution trajectory analysis This hybrid approach combines numerical efficiency for complex systems with symbolic precision for analytically tractable cases, providing comprehensive solution strategies for algebraic differential equations.