Ordinary Differential Equations

An ordinary differential equation (ODE) is a function equation that involves some ordinary (rather than partial) derivatives. Our goal is frequently to solve an Ordinary Differential Equation, that is, to figure out which function or functions fulfil the equation.

How can you find the function itself if you know what the derivative of a function is? You must identify the anti-derivative, which means you must integrate. For instance, if you’re given

dx/dt (t)=  cos t

So, you would aim to answer the x(t) function. As we know, sin t is the anti-derivative of cos t,  x(t) must also be in t. Except for one thing, if we only see the derivative, there is always an arbitrary constant “+c” that we cannot determine. As a result, we can deduce from the equation above that

x(t)= sin t + C – for some arbitrary constant C. You can verify that x(t) satisfies the equation dx/dt=cos t.

Solving an Ordinary Differential Equation is more complex than simple integration in most cases. Nonetheless, because we must proceed from derivative to function, the essential premise is always integration. Determining what integration we need to perform is usually the most challenging step.

Defining the Order of Given Ordinary Differential Equation:

An “Ordinary Differential Equation,” or Ordinary Differential Equation, is a mathematical equation with only one independent variable and one or more derivatives about the variable. 

y’,y”, ….yn   with respect to x.

Now, what is meant by the “Order” ? 

The order of ordinary differential equations is the order of the highest derivative. The following is the general form of an n-th order Ordinary Differential Equation: 

F(x, y,y’,….,yn ) = 0

Note that, y’ can be either dy/dx or dy/dt and yn can be either dny/dxn or dny/dtn.

An ordinary differential equations of nth order is linear if it is written in the form;

a0(x)yn + a1(x)yn-1 +…..+ an(x)y = r(x)

The function aj(x), 0 ≤ j ≤ n, is called the linear equation’s coefficients. 

Types

The ordinary differential equation is further classified into three types. They are:

• Autonomous Ordinary Differential Equation
• Linear Ordinary Differential Equation
• Non-linear Ordinary Differential Equation

Autonomous Ordinary Differential Equation

An autonomous ordinary differential equation is an equation of the form

dy/dt=f(y).

Assuming that t indicates time, these differential equations depict the rate of change by/dt of the function y(t) is given by a rule. The rule says that if the current value is y, then the rate of change is f(y).

The differential equation is autonomous because the rule doesn’t depend on the time t. It is solely dependent on the current value of the variable y.

Linear Ordinary Differential Equation

A linear ordinary differential equation  is linear in both x(t)x(t) and its first order derivative  dx/dt (t). An example of such a linear Ordinary Differential Equationis

dx/dt+t3x(t)=cost

Even though this Ordinary Differential Equation is nonlinear in the independent variable t, it is still regarded as a linear Ordinary Differential Equation because we are only interested in the equation’s dependency on x and its derivative. Int, we can easily manage nonlinearities. Such nonlinearities may result in integrals that cannot be computed analytically. Still, if we can define x(t) as an expression comprising only integrals of functions of t, we can solve the differential equation

Non-linear Ordinary Differential Equations

If the differential equations cannot be written in the form of linear combinations of the derivatives of y, then it is known as a non-linear ordinary differential equation.

When an Ordinary Differential Equation’s time integration yields a nonlinear algebraic equation, we must generally solve it by constructing a series of linear equations and hoping that the solutions of these linear equations converge to the nonlinear algebraic equations’ desired solution. Because only a few nonlinear systems can be solved explicitly, the answer is usually approximated using a numerical approach.

Conclusion:

Ordinary Differential Equations have a wide range of applications and can forecast the world around us. Biology, economics, physics, chemistry, and engineering are just a few of the fields that employ it. It aids in the forecasting of exponential growth and decline, as well as population and species growth. Ordinary Differential Equations can be used for a variety of purposes, including but not limited to: modelling illness progression, the movement of electricity, the pendulum’s motion as well as waves and Newton’s second law of motion and the Law of Cooling both use this term.