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The following article was published in our article directory on March 7, 2013.
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Understanding Differential Equations

Article Category: Advice

Author Name: Timur Karipov

Mathematical equations for unknown functions of either one or more variables relating to the values of the functions themselves, including the derivatives coming from various orders are called differential equations They execute vital roles in the realm of physics, engineering, economics, including other applicable disciplines. These are equations involving derivatives of certain functions.

Whenever partial derivatives will be involved, the resulting equations are then called partial differential equations. If ever only the ordinary derivatives become present in the equation, then it will then be referred to as an ordinary differential equation. Whatever the case is, a differential equation plays a useful role in applied physics, math, and engineering, with much of the mathematical or numerical machineries developed following its established perimeters.

A differential equation may also come in quite handy in so many areas within science and technology, especially in cases where deterministic relations that involved continuously varying quantities (modeled of course by the functions) and the subsequent rate of change in real time and space (known as derivatives) is made known. Classical mechanics paints a good picture of this; the motion of bodies is described by the given velocity and position as time values becomes varied.

It must be noted that Newton's laws acknowledge a body (given its position, acceleration, velocity and the other forces acting on it) to express such variables in dynamic ways, as would in the case of a differential equation intended for unknown positions acted upon by the body due to the function of time. This particular differential equation (also referred to as the equation of motion) can be solved in an explicit manner.

The laws that Nature had upon us are also expressed as differential equations. Both engineers and scientists must learn to appreciate the modeling of the world in such thinking and equations, and also solving and interpreting such equations.

A good example for modeling real world problems through the idea of a differential equation would be the determining the velocity of a certain ball that falls freely through the air, with air resistance and gravity as the only consideration involved. The acceleration of the ball into the ground becomes the action because of the gravity involved without the deceleration factor because of the interference of air resistance. Gravity must be considered a constant here, and the instance of air resistance will be considered proportional to the velocity of the ball. What this means is that the acceleration of the ball, a derivative of the velocity involved, also depends on the velocity itself (where the velocity is also dependent on time). Interpreting the velocity as a time function will involve solving differential equations.

A differential equation is also mathematically studied upon using several perspectives that are mainly concerned with deriving the solutions - a set of the functions that satisfies the equations. You must understand that only a simple differential equation admits solutions offered by explicit formulas. Some properties of the solutions of given equations however can be determined without relying on the exact form. If self-contained formulas intended for the solutions are not made available, then it is expected that solutions can be numerically approximated with the help of computer systems.

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