General Solution for Differential Equation

A differential equation solution is a relationship between the variables (independent and dependent) that is free of derivatives of any order. In simpler words, a differential equation is a mathematical equation that connects one or even more undefined values with their derivatives. Also, differential equations will have infinite solutions for which it has been named as general solution. 

In applications, features are used to represent measured data, deviations are used to describe their rates of change.  Thus, some tips or approaches can help to solve differential equations. 

Differential Equations Types

A differential equation’s outcome is a solution achieved by allocating specified values to the indeterminate constants. Depending on the mathematical problem, the criteria for computing the numbers of the indeterminate constants can be specified in the form of an Initial-Value Problem or Process Parameters. Thereby, using initial value will help in finding not more than one solution. Besides, specific solutions are those which leave no constants that are unknown. There are methods or types to solve differential equations. 

1. Ordinary Differential Equations: An ordinary differential equation is a mathematical expression that involves derivatives of the dependent variable concerning to explanatory variables.
2. Linear Differential Equations (LDEs) are equations that contain a variable, its derivative, and a few other functions.
3. Differential Equations that Aren’t Linear: Nonlinear differential equations are a type of equation that contains nonlinear terms.
4. Non-Homogeneous Differential Equations- To solve differential equations of nonhomogeneous second order differential equations, it is important to find a general solution for the complementary equation after which specific solutions for non- homogenous are found. 
5. Partial Differential Equations (PDEs) are equations containing unknown multivariable and their partial derivatives.

It is important to remember that not every differential equation can be solved, thus, the existence question should be answered first before starting the calculations. 

Application

Furthermore, differential equations solution plays an important role in many areas, including technology, physics, economics, and biology because such relationships are widespread. Some of which have been listed below. 

• Differential Equations in Real Life – Differential Equations are used in a number of different disciplines, including engineering, mathematics, and science.
• They are employed in medical research to imitate cancer growth and illness dissemination in the body.
• Calculate the population levels and decline of any country.
• It’s utilized to figure out Newton’s 2nd Law of Motion and Newton’s Law of Cooling.
• A differential equation can be used to calculate the mobility of electricity.
• Calculate the body’s glucose absorption.
• This equation can also be used to describe the motion of a pendulum or ocean waves.
• It is utilized in engineering courses such as Strength of Materials to calculate bridge, beam, and column calculations, Thermodynamics, and so on.

Things to Keep in Mind

To solve differential solutions, some of the points below could be referred to. 

• The only independent variable in Ordinary Differential Equations is ‘y.’
• There are two or more independent variables in partial differential equations.
• Before attempting to solve any differential equation, it is necessary to consider the order and degree of the problem.
• Make sure you don’t mix up order and degree, as they’re the two most important criteria for solving differential equations.
• When the original equation is in the form of a polynomial equation in derivatives such as y’,y”,y”‘, the level of a differential equation is defined as the ‘rank of the highest order.’

Steps to solve differential equation

• Substitute uv for y
• Add up the parts that involve v.
• Set the value of the v term to zero 
• To find u, use the separation of variables method.
• Replace u in the equation we got in step 2 with u.
• To find v, solve it

Conclusion

Differential equations are challenging to use, although they are very helpful for solving complex issues. To solve differential equations, it is important for converting it into a simpler one without using the differential elements. This will help in making calculations, graphs and predictions. It should be remembered that the pace (rate) of change of a function at a particular point is defined by its derivatives. A differential equation’s order and degree should always be relatively positive (+)