ODEs where the dependent variable is not proportional to its derivatives and the equation is not homogeneous.
Homogeneous Equations: An ODE is said to be homogeneous if all of its terms are of the same degree.
Integrating Factors: Integrating factors are used to simplify the process of solving non-homogeneous equations.
Variation of Parameters: In this method, one assumes that the solution of the non-homogeneous equation is a linear combination of the solutions of the corresponding homogeneous equation.
Constant Coefficients: Constant coefficients refer to the fact that the coefficients of the ODE remain the same throughout the equation.
Undetermined Coefficients: In this method, one attempts to find a particular solution to the non-homogeneous equation based on its form.
Laplace Transforms: Laplace transforms can be used to solve non-homogeneous equations, although they are not typically used as the primary method.
Power Series: Power series can be used to solve non-homogeneous equations by expanding the solution around a point.
Green's Functions: This method involves solving the homogeneous problem and then using the Green's function to solve the non-homogeneous problem.
Boundary Value Problems: In boundary value problems, the solution to the equation must satisfy certain conditions at the boundaries.
Initial Value Problems: In initial value problems, the solution to the equation is determined by giving the value of the solution and its derivative at a certain point.
Linear Differential equations: These are ODEs that are linear in both their dependent variable and its derivatives.
Non-linear Differential equations: These are ODEs that are not linear in the dependent variable and/or its derivatives.
Separable Differential equations: These are ODEs in which the dependent variable and its derivative(s) can be separated and solved separately.
Exact Differential equations: These are ODEs that can be written as the derivative of a function of both the dependent variable and an independent variable.
Bernoulli Differential equations: These are ODEs that can be transformed into a linear form by substituting the dependent variable with a related function.
Riccati Differential equations: These are Non-linear ODEs in which the dependent variable can be transformed into a linear form using a substitution technique.
Legendre Differential equations: These are ODEs that arise in mathematical physics and are characterized by their dependence on a Legendre polynomial.
Bessel Differential equations: These are ODEs that arise in mathematical physics and involve a Bessel function as a solution.
Laplace Differential equations: These are ODEs that can be solved using the Laplace transform.
Fourier Differential equations: These are ODEs that can be solved using the Fourier transform.