Differential equations

Differential equations were developed by Newton (1642-1727) and Leibniz (1646-1716). Their idea was to try to solve a problem by considering infinitely small changes in time. This leads to differential equations. The fact that one can reformulate these problems as differential equations has numerous advantages.  Differential equations are widely used in physics, engineering, weather forecasting, and even economics. Differential Equations is a field of study that allows one to understand and predict the changing state of a system. . Many of the systems we see around us are dynamic or increasing in complexity over time. 

In some cases, the differential equation can be expressed as an equation of motion.

Differential equations are often used to model real world problems such as the velocity of a ball falling and considering only gravity and air resistance. The ball’s acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance and is calculated by solving a differential equation that depends on time.

Definition

A differential equation is one that has at least one derivative of an unknown function, which can be either an ordinary or partial derivative. Assume the rate of change of a function y with respect to x is inversely proportional to y, which we may express as dy/dx = k/y.

A differential equation in calculus is an equation involving the derivative (derivatives) of the dependent variable regarding the independent variable (variables). The derivative is just a rate of change, and the differential equation allows us to establish a relationship between the changing quantity and the change in another quantity. Be a function y=f(x), where y is a dependent variable, f is an unknown function, and x is an independent variable.

Order of Differential equation

Differential equations are categorized by the order of their derivative, determined by the presence of higher derivatives in the equation. A differential equation with only first derivatives is a first-order differential equation, a differential equation with second derivatives is a second-order differential equation, and so on. 

Differential equations can be divided into several types. These types include whether the equation is ordinary or partial, linear, or nonlinear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations that can be very useful in specific contexts.

Types of differential equation

• Ordinary differential equations
• Partial differential equations

Ordinary differential equations

The ODE or “Ordinary Differential Equation” is an equation that contains only 1 independent variable and more than one of its derivatives with respect to the variable. The  given Ordinary differential equations can be used in complicated math that uses 1 variable, x, and some constants such as y.

Linear differential equations are the most important form of differential equation and  the solutions may often be expressed in the terms of integrals.

Differential equations are notoriously hard to solve. We usually must use computational methods to solve them.

F(x) = y = dy/dx

Partial differential equations:-

With a partial differential equation, you can formulate problems that involve several variables. They can be solved in closed form and used to create a computer model.

Different physical phenomena can be modelled by partial differential equations, including sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, and quantum mechanics. The content of these PDEs are similar to those of ordinary differential equations; they just cover multiple dimensions.

{dz}/{dx}+{dz}/{dy}= 2z is a PDE of first order

The importance of Non-linear differential equations in order

Nonlinear differential equations are typically more complicated and have more potential in terms of chaos, than linear differential equations. If a differential equation is correctly formulated, then one would be expecting it to have solutions.

Linear differential equation:-

A linear differential equation is accurate under a condition of small amplitude oscillations. For example, the harmonic oscillator equations approximate the pendulum equations that can be used when there is limited movement in their amplitude.

System of Differential Equation

Let us now look at how to solve systems of differential equations. For the purposes of this study, we shall limit ourselves to systems of two linear differential equations, however many of the concepts will apply to larger systems of linear differential equations. We also look at sketch phase planes/portraits for two-differential-equation systems. In addition, we include brief discussions on how to solve systems using Laplace transforms and some modelling that results in systems of differential equations.