DefinitionOrdinary Differential Equations in which the order of the derivative of y is no more than 1 (i.e. the highest derivative is the first derivative ) Depending on the properties and coefficients involved, first-order ODEs can be of the separable, linear, exact, or homogeneous type, as well as Bernoulli equations. We can use first-order ODEs to model rates of change, growth, decay phenomena; like in physics, engineering, and mathematical modeling. Analytically it is solved with the help of techniques involving separation of variables, integrating factors, and variation of parameters, and numerically it is solved using Eulers method and Runge-Kutta methods. For example, the first-order ODE dxdy=x2+y is non-linear only with respect to the solution z(x,y) to produce exponential growth or decay with a quadratic source term, and can be solved using integrating factors to convert this DE into a linear relation or the solution can be approximate using numerical methods.
