Solutions of a Differential Equation

Differential equations are those equations that have one or more than one derivative of a function. The solution of a differential equation, say y = f (x), satisfies a differential equation by substituting the value of f and its derivatives in the equation. The most common differential equation that everybody knows is Newton’s second law of motion which is applicable in our daily lives. Therefore, the solutions of a differential equation are applied in the case of population growth, heat flow, planetary movement, electrodynamics, economic systems, etc. 

There are various techniques such as – direct solution, using graphs, or computer solution; that can be applied to find the solution of a differential equation. 

An equation consisting of an unknown function, say y = f (x), and its derivatives are referred to as a differential equation. 

Information on how quantities change and why do they change is provided on solving these equations, known as solutions of the differential equation.

To find the general solution of a differential equation, first, we have to find the solutions of the first order and second order. 

Suppose you want to solve the first-order differential. In that case, a variable should be kept in a separable method. We have to mention at least one arbitrary constant before we can proceed to perform any type of integration. 

Therefore, we can consider that the solution of the first-order differential equation has at least one of the Unique or arbitrary constants fixed after simplification.

A particular solution of a differential equation – 

A particular solution of a differential equation cannot be removed from the general solution since it is allocated specific values at random constants.

Homogeneous differential equation 

A homogeneous solution of the differential equation is an equation containing a homogeneous function. 

• What is a homogeneous function? 

The homogeneous functions of a homogeneous differential equation do not have a constant term. The linear differential equation solution is possible if one can remove the constant term from the differential equation and transform it into a homogeneous differential equation that does not have any X and Y variables or any special logarithmic or trigonometric function. 

• How to solve a homogeneous differential equation? 

 The general solution of a linear equation can be very easy if solved following certain steps. This general solution can be obtained by the integration of the given differential equation. 

First, to solve a homogeneous differential equation, it has to be expressed in the form of DY by DX, which equals to the function of X and Y. When a first-order differential equation can be expressed in the form: dy / dx = f (y / x), it is termed as a homogeneous differential equation.

This is to be solved by separating the variable and derivatives of the particular equation by substituting the value y/x by another new variable, say v. From the integration of the given differential equation, the general solution of a homogeneous differential equation is obtained.

Second-order differential equation 

In the second-order differential equation, the unit is removed from the first-order differential equation to the second-order differential equation. 

They play quite an important role in the Physical Sciences since their applications have been applied in describing mechanical systems, Wave Motion, electrical current, and Quantum phenomena.

When someone is working with a differential equation, their goal is usually to find a solution for the second-order differential equations. 

The general form of a homogenous second-order differential equation is a (d2y / dx2) + b (dy / dx) + cy = 0. Here, x and y are independent variables whereas a, b, and c are constants.

First-order differential equation

DY defines the first-order differential equation by DX; this differentiation is used to express different types of relations between the functions and their derivatives; this is directly applied.

In Physics and Chemistry, it is a technique used to determine the domains and their functions in different projects.

Two methods that can be used to find out the first-order differential equation are – the integrating factor and the method of variation of constants.

The different applications of a first-order differential equation are – used in Newton’s law of cooling, growth and decaying, falling body problems, electrical circuits, etc.

The first order of differential equations possesses different properties. It should not have any transaction functions as logarithm for trigonometric functions, and the products of Y and any office derivatives are not present.

Conclusion:

Differential equations are equations that have at least one functional derivative and the value of the derivatives are substituted to find a relevant answer, like Newton’s second law of motion which is a classic example of differential equations. The differential equations are categorized into types each suitable for a particular type of solution. In the first order differential equation at least one unique constant is fixed after simplification. While the homogeneous differential equation does not have a constant term a linear equation would be possible only if the constant term is removed. In the second order differential equation a unit is removed from the first order differential equation to the second order differential equation.