Math Assignment Help >> Differential Equations

A differential equation is a equation or mathematical equation for given an unknown function of one or more variables that relates the function’s value itself and its derivatives in various orders. Differential equations play a prominent role in physics, engineering economics and other disciplines.

Differential equations arise in several areas of science & technology, mainly, whenever a deterministic relation including some continuously varying quantities that is modeled by functions and their rates of change in space or time which is expressed as derivatives is known or postulated. It is illustrated in classical mechanics, in which the motion of a body is mentioned by its position and velocity with the varying time. Newton's laws allow one to relate the position, acceleration, velocity and other forces acting on the body and state this relation as a differential equation for the unknown position of the body as a time function. In some of cases, this differential equation which is called an equation of motion may be solved explicitly.

A real world problem for example modeling using differential equations is determination of the velocity of a ball falling through the air, considering only air resistance and gravity. The ball's acceleration towards the ground is the acceleration caused by gravity minus the deceleration caused by air resistance. Gravity remains constant but air resistance can be modelled as proportional to the velocity of ball. This means the acceleration of ball, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time includes solving a differential equation.

Differential equations are mathematically learned from several different perspectives, mostly concerned with its solutions—the set of functions that satisfy the equation. Only the simplest differential equations concerns solutions given by explicit formulas; however, some differential equation and properties of solutions can be determined without finding their exact form. If a self-determined formula for the solution is not available, the solution can be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to show solutions with a given degree of accuracy.

The learning of differential equations is a wide field in physics, applied mathematics, meteorology, and engineering. These disciplines are concerned with the properties of differential equations of various types. Pure mathematics mainly focuses on the uniqueness and existence of solutions, in applied mathematics emphasizes the rigorous justification methods with approximating solutions. Differential equations take an important role in modelling virtually every physical, biological process or technical, from celestial motion, to interactions between neurons, to bridge design. Differential equations for example those used to solve real-life problems may not necessarily be directly solvable or mark able.

In differential equations theory is quite developed and the methods used to learn them vary significantly with the type of the equation.

- An ordinary differential equation - ODE is a differential equation in which the unknown function which is known as the dependent variable is called function of a single independent variable. In the simple form, the unknown function is a complex valued or real function, but more commonly, it can be matrix-valued or vector-valued: this corresponds to consider a system of ordinary differential equations for a given single function. Ordinary differential equations are classified according to the order of the highest derivative with the dependent variable with respect to the independent variable appearing in the given equation. The important cases for applications are known as first-order and second-order differential equations. In the classical literature distinction is made between differential equations explicitly solved with differential equations in an implicit form and respect to the highest derivative.

- A partial differential equation - PDE is a differential equation where the unknown function is a function of multiple independent variables or the equation involves its partial derivatives. The order is defined same to the case of ordinary differential equations, but further classification into hyperbolic, elliptic, and parabolic equations, especially for second-order linear equations, is of utmost importance.

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