MATH 133A Ordinary Differential Equations
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In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more independent variables and the derivative of these functions. Unlike the term partial differential equation, this term is used in comparison with partial differential equations, including several independent variables.
In ordinary differential equations, linear differential equations play an important role for several reasons. Most of the essential and special functions found in physics and applied mathematics are linear differential equations (see Complete functions). When modeling physical phenomena using nonlinear equations, linear differential equations are usually approximated to solve the problem. The few nonlinear ODEs that can be solved explicitly can generally be solved by transforming the equations into equivalent linear ODEs (see, for example, Riccati equations).
Some ODEs can be solved explicitly based on well-known functions and integrals. If this is not possible, the Taylor series equation used to calculate the solution may be helpful. For applied problems, numerical methods of ordinary differential equations can give an approximate value for the solution.
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