Ordinary differential equation
Differential equations

A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

where a 0 (x) {\displaystyle a_{0}(x)},..., a n (x) {\displaystyle a_{n}(x)} and b (x) {\displaystyle b(x)} are arbitrary differentiable functions that do not need to be linear, and y ′, …, y (n) {\displaystyle y',\ldots,y^{(n)}} are the successive derivatives of the unknown function y of the variable x.

Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation).

Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.
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