Differential Equations

Introduction to Differential Equations, Initial Value Problems, Variable Separable method, and Homogeneous Differential Equations.

Solving First-Order DEs using Exactness, Integrating Factors, and Linear Methods (including Bernoulli's Equation).

Practical applications including population growth, decay, Newton's Law of Cooling, mixing problems, electrical circuits, and falling bodies.

Solving nth-order linear homogeneous differential equations with constant coefficients and Cauchy-Euler equations.

Solving non-homogeneous linear differential equations using Undetermined Coefficients and Variation of Parameters.

Practical applications in mechanical and electrical systems, including Spring-Mass Systems, Pendulums, and RLC Circuits.

Solving systems of linear differential equations using elimination, matrix methods (eigenvalues), and phase portrait analysis.

Solving initial value problems using the Laplace Transform method, partial fraction decomposition, step functions, and Dirac delta.

Using power series methods, including Radius of Convergence and the Frobenius Method, to solve differential equations with variable coefficients.

Solving differential equations using numerical approximations, including Euler's method, Runge-Kutta methods, and error analysis.

Introduction to PDEs, classification, Separation of Variables, and solving the Heat, Wave, and Laplace equations.