First Order Differential Equations

The order of any differential equation is the highest derivative of the dependent variable to the independent variable. It helps find the degree of differential equations and solve many differential equations. 

The differential equation with the highest degree is a first order differential equation. All the linear equations in the form of derivatives are in first order. We can present the first order differential equations as: 

dydx=fx,y=y’

History 

Gottfried Wilhelm Leibnitz (1646-1716) first stated differential equations on 11th November 1675, identifying it as y.dy=12y2 introducing both the symbols for the first time. 

Leibnitz discovered the ‘method of separation of variables’ in 1691. He introduced the ‘method of solving a linear differential equation of the first order’ a year later. 

Types of first order differential equation

There are different types of differential equations of the first order:

• Linear differential equation: A first order differential equation, y‘=f(x,y), is a linear equation if it can be written as y‘+pxy=q(x), where p and q are the continuous functions on some specific interval I. The differential equation that doesn’t seem linear is known as a non-linear differential equation.
• Separable equations: A first order differential equation, y‘=f(x,y), is a separable equation if the variables can be written separately as fx,y=px.q(y), here p and q being the continuous functions on some specific interval I.
• Homogeneous equations: A first order differential equation, y‘=f(x,y), is homogeneous if f(kx,ky=f(x,y) for all k≠0. If we change the variable =vx,dydx=v+x.dvdx, it transforms the homogeneous equation into separable equation. 

Ways to solve a first order differential equation

There is no general formula for calculating the solution of a general differential equation. So, we will discuss two specific ways to solve different types of first order differential equations. For solving linear differential equations, we can use the integrating factor method. For the calculation of separable differential equations, the integral curves method can be used, and homogeneous differential equations can be solved by reducing them into separable equation forms.

• Integrating factor: In this method, there is a multiplication by the factor, known as integrating factor, which enables us to write the left side of the equation as the derivative through which we get the general solution. The crucial step in solving is: y‘+pxy=q(x), is a multiplication by eh(x), where hx=px.dx.

• Integral Curves: These are the curves that are tangent to everywhere on the direction field, and each integral curve represents the solution of a differential equation. In general form, it can be said that the integral curves define y as a function of x implicitly. The set of integral curves forms at the centre of origin, the family of circles.

Applications of a first order differential equation

Let’s discuss some applications of first order differential equations-

1. Modelling population growth: Population growth can be modelled with the law of exponential change as

 dpdt=kp,p0=p0 

Where P is the population at time t, k>0 is a constant growth rate, and p0 is the population size at time t=0. But in practical life, the rate of increase in population does not continue its proportionality with the size of the population.

1. Orthogonal trajectories: An orthogonal trajectory for a set of curves is a curve intersecting each curve of the set at right angles. The mutually orthogonal systems of curves are of specific importance in problems related to electric potential, hydrodynamics and heat flow.

2. Resistance proportional to velocity: In some cases, not all, the resistance experienced by the moving object is proportional to the object’s velocity. According to Newton’s second law of motion, the resisting force opposing the motion is

Force=mmassdvdt(acceleration) 

If the resisting force is proportional to velocity, we get 

 m.dvdt= -kv

dvdt=-km.v

This represents a separable differential equation. The solution to the equation with initial condition v=v0 at t = 0:

v=v0.e–kmt

1. Exponential growth and decay: From the observations, it can be concluded that the rate of decay of a radioactive material at the time (t) is always proportional to the amount of material present at the time (t). One important property of a radioactive material is the length of time it takes to decay to one-half of the initial amount.

Conclusion

Differential equations of the highest degree one are the first order differential equations. Here, the product of any derivative of y and y itself is absent, and it doesn’t contain any transcendental functions. There are three types of first order differential equations. There are two ways to solve differential equations. They only have the first derivative, such as dydx, where x and y are the two variables. It can be used in many ways, such as for modelling population growth, for studying exponential growth and decay, in Newton’s cooling law etc.