A differential equation is a mathematical relationship which contains one or more derivatives of a function. The solution of the differential equation is the original function. These equations are often used in scientific applications, particularly in physics and engineering, because they express rates of change in quantities of interest such as position, temperature, and other physical parameters. They also can be considered an extension of the basic study of calculus.
Differential equations are classified by several features. The order of a differential equation is given by the largest number of derivatives of a function featured in that equation - thus, an equation featuring third-derivatives of a function would be a third-order differential equation, but an equation featuring both third- and fourth-derivatives would be a fourth-order differential equation. Most differential equations used in practical applications are first- or second-order differential equations, as there are very few third-order rates of change that are useful in physical settings. However, the higher order equations do appear and may be useful for obscure and theoretical studies.
Differential equations can also be divided into ordinary and partial categories. Ordinary differential equations only feature ordinary derivatives, whereas partial differential equations involve partial derivatives. In the case of partial differential equations, the partial derivatives may be in more than one variable. Often, differential equations will appear with partial derivatives in all spatial directions and time.
Linearity is another way to classify differential equations. If a differential equation is linear, it does not feature the product or any other non-additive combination of different orders of derivative - that is, a linear differential equation will not feature a term like yy', but y + y' is acceptably linear. Nonlinear equations are generally much harder to solve because they cannot be separated into differential and non-differential components as easily.
Systems of differential equations with more than one unknown function can occur. These contain more than one unknown function in differential form, at any level of derivative. As with systems of linear equations, systems of differential equations must be solved as an entire unit, combining and eliminating functions instead of algebraic variables.
Methods of solving differential equations vary according to the type of equation being solved. Some simple differential equations may be separated into a differential and a non-differential component. Both sides of the equation are then integrated to find the solution. For some differential equations, a simpler related equation can be solved with a correction term for the more complex case. All differential equations may be solved with a polynomial series. The polynomial solution is known as Frobenius' Method. For other differential equations that are to be used in a practical settings, numerical solutions may be provided, in which the behavior of the equation is examined in small incremental steps. The advent of the personal computer in the workplace makes this solution method much more feasible than it was when all calculations were done by humans directly. The numerical calculations are often used when the equation has been separated and the resultant integral is intractable to solve in closed form.
To get exact solutions to differential equations, one must have some information about the equation's specific behavior. Usually there is a family of solutions which would solve the differential equation, varying by one or several constants, depending upon the order of the equation. Boundary value problems tell the person solving the equation what behavior the solution function will have at more than one variable location. Initial value problems tell the solver the behavior of all the derivatives of the function at one variable location. In either case, the number of specified parameters must be equal to the order of the equation. The family of equations is then examined algebraically to determine the values of the constants in the solution.
There are many famous differential equations whose solution is already known. In this case, it only remains to recognize the equation one has as a form of the known equation. The motion of waves in a medium, simple oscillatory motion, and the motion of a struck drumhead are all expressed in familiar, solved differential equations, and the application of the equations to these situations only requires a knowledge of boundary or initial value conditions. Most physics problems involve the solution of a differential equation at some point. They are essential to classical and quantum physics alike, as well as to some chemical rate reactions and many formulas used in chemical, electrical, mechanical, and environmental engineering.
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