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In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations.
Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.
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 ...
Let y (n) (x) be the nth derivative of the unknown function y(x).Then a Cauchy–Euler equation of order n has the form () + () + + =. The substitution = (that is, = (); for <, in which one might replace all instances of by | |, extending the solution's domain to {}) can be used to reduce this equation to a linear differential equation with constant coefficients.
For instance, Legendre's differential equation can be shown to be a special case of the hypergeometric differential equation. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of Legendre's differential equation, after making the necessary substitutions.
A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation. A solution of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a vector space. In the ordinary case, this vector space has ...
The solutions to an exact differential equation are then given by (, ()) = and the problem reduces to finding ψ ( x , y ) {\displaystyle \psi (x,y)} . This can be done by integrating the two expressions M ( x , y ) d x {\displaystyle M(x,y)\,dx} and N ( x , y ) d y {\displaystyle N(x,y)\,dy} and then writing down each term in the resulting ...
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