The elements of higher mathematics Differential Equations Lecture IV The elements of higher mathematics Differential Equations

Lecture questions Differential equation definition Ordinary and partial differential equations The order of differential equation General and particular solutions of differential equation Particular solution, supplementary conditions Separable differential equation

Differential Equation A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders or differentials.

Universality of mathematical description The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, engineering, chemistry, biology and economics. All of these disciplines are concerned with differential equations of various types. They are used to model the behavior of complex systems. Many fundamental laws can be formulated as differential equations. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons.

Ordinary and partial differential equations An ordinary differential equation (or ODE) is a relation that contains functions of a single independent variable, and one or more of their derivatives with respect to that variable or differentials. A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives or partial differentials.

Order The order of differential equation is defined as the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. For example : is the first-order ODE is the second-order ODE

To solve the equation means to determine the unknown function which will turn the equation into an identity upon substitution. (Solving a differential equation means finding a function that satisfies the given differential equation.) Solving a differential equation always involves one or more integration steps.

General and Particular Solutions The general solution of the differential equation y′ = 2 x is y = x2 + c, where c is any arbitrary constant. Note that there are actually infinitely many particular solutions, such as y = x2 + 1, y = x2 − 7, or y = x2 + π, since any constant c may be chosen. Geometrically, the differential equation y′ = 2 x says that at each point ( x, y) on some curve y = y( x), the slope is equal to 2 x. The solution obtained for the differential equation shows that this property is satisfied by any member of the family of curves y = x2 + c (any only by such curves).

Particular solutions graphs

Particular solution, supplementary conditions If one particular solution (or integral curve) is desired, the differential equation is appended with one or more supplementary conditions. These additional conditions uniquely specify the value of the arbitrary constant or constants in the general solution.

Separable differential equation is the differential equation of the form

To solve a separable differential equation perform the following steps: 1) Separate variables x and y in different sides of equation together with differentials dx and dy. Move any term into the left side and divide both parts by to obtain the equation where the variables are separated:

2) Integrate the equation: 3) And obtain a general solution: once we find antiderivatives, we put the +C on just one side (because two additive constants can be absorbed into one). 4) If you are given the initial (or boundary) conditions use them to find the particular solution.

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