solution through integration. In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in summation), and the notation just means to integrate with respect to , then after the integration substitute , without adding constants (explicitly stated). Separable equations General first-order equations General second-order equations Linear to the nth order equations The guessing method When all other methods for solving an ODE fail, or in the cases where we have some intuition about what the solution to a DE might look like, it is sometimes possible to solve a DE simply by guessing the solution and validating it is correct. To use this method, we simply guess a solution to the differential equation, and then plug the solution into the differential equation to validate if it satisfies the equation. If it does then we have a particular solution to the DE, otherwise we start over again and try another guess. For instance we could guess that the solution to a DE has the form: since this is a very common solution that physically behaves in a sinusoidal way. In the case of a first order ODE that is non-homogeneous we need to first find a DE solution to the homogeneous portion of the DE, otherwise known as the characteristic equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions together to obtain the total solution to the ODE, that is: Software for ODE solving Maxima, an open-source computer algebra system. COPASI, a free (Artistic License 2.0) software package for the integration and analysis of ODEs. MATLAB, a technical computing application (MATrix LABoratory) GNU Octave, a high-level language, primarily intended for numerical computations. Scilab, an open source application for numerical computation. Maple, a proprietary application for symbolic calculations. Mathematica, a proprietary application primarily intended for symbolic calculations. SymPy, a Python package that can solve ODEs symbolically Julia (programming language), a high-level language primarily intended for numerical computations. SageMath, an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics. SciPy, a Python package that includes an ODE integration module. Chebfun, an open-source package, written in MATLAB, for computing with functions to 15-digit accuracy. GNU R, an open source computational environment primarily intended for statistics, which includes packages for ODE solving. See also Boundary value problem Examples