Differential Equations can be a difficult topic because it involves a number of complex concepts. Understanding the concepts and properties of differential equations plays an important role in understanding mathematics, science and engineering. Differential Equation plays an important role in physics, economics, engineering and other disciplines. Students seek help for their differential Equation assignment help and need assignment help provides differential equation assignment help to the students who face challenges in writing their differential equation assignments with the unmatched services at low prices.
WHAT IS DIFFERENTIAL EQUATION ?
A differential equation contains one or more terms involving derivatives of one variable with respect to another variable.
For example, dy/dx=2x
Here, y is the dependent variable and x is the independent variable.
The solutions of differential equations are not numbers, they are functions and represents the relationship between continuously varying quantity and its rate of change.
A differential equation simply states how a rate of change in one variable is related to other variables.
EXAMPLE OF DIFFERENTIAL EQUATIONS
For a better understanding of differential equations. Let us understand it with the help of an example :
Let us consider a simple equation :
x^{2}+ 2x+ 1 = 0
It has a solution x=-1
But in case of a differential equation, the solution cannot be a single value, but it will be a function. And to solve a differential equation our aim is to find a function whose derivatives meet the differential equation over a long time.
For example :
x'' + 2x' +x= 0 (1)
Now, this is a differential equation, where we have to find a function x(t). And the general solution for this equation will be
x(t) =ae^{−t }+ bte^{−t} (2)
The value for e is 2.71828 and a, b are constants. Now let us find out the first and second derivative of the equation.
The first order derivative will be :
x'(t) = (b−a) e^{−t}−bte^{−t} (3)
The second order derivative will be :
x''(t) = (a− 2b) e^{−t}+bte^{−t} (4)
Now using the equation number (2), (3) and (4) on the right side of the differential equation (1)
x'' + 2x' +x= ((a− 2b) e^{−}t+bte^{−}t) + 2 ((b−a) e^{−}t−bte^{−}t) + (ae^{−}t+bte^{−}t)
=(a− 2b + 2b− 2a + a) e^{−}t+ (b− 2b + b)te^{−t}
= 0
This solution is called the general solution.
CLASSIFICATION OF DIFFERENTIAL EQUATIONS
Some of the major classifications of differential equations are :
FIRST ORDER, SECOND ORDER etc
The highest derivative in a differential equation is the order of differential equation, where a’ is the first derivative, a’’ is the second derivative.
For example : a'' + 2a' +a= 0 is second-order.
LINEAR VS NON-LINEAR
Linear means that the variable appears with a power of one. So a is linear, a2 is a non-linear and the functions like sin (x) is non-linear.’
Some of the examples of linear vs non-linear are
a' + 1/a= 0 is non-linear because 1/ais not a first power
a' +a^{2}= 0 is non-linear becausea2is not a first power
a'' + cos (a) = 0 is non-linear because cos (a) is not a first power
a a' = 1 is non-linear becausea' is not multiplied by a constant
HOMOGENEOUS VS NON-HOMOGENEOUS
The term that involves only time in an equation is the non-homogeneous part of the equation.
For example :
a'' + 2a' +a= 0 is homogeneous
a'' + 2a' +a= cos (t) is non-homogeneous
a' +t^{2} a= 0 is homogeneous
a' +t^{2} a=t+t^{2} is non-homogeneous
NUMERICAL VS ANALYTICAL SOLUTIONS
When you know about the behavior of model under different circumstances is known as analytic solution. It also referred as a closed form solution.
APPLICATIONS OF DIFFERENTIAL EQUATIONS
Differential Equation is very important in biological, physical and technical process (bridge design, celestial motion and interaction between neurons). Differential equations have open form solutions.
The applications of differential equation in real life are ; chemistry, biology, physics and the other areas of natural sciences and economics and engineering.
FEW EXAMPLES OF ITS APPLICATIONS ARE :
CHEMISTRY : The rate law in a chemical reaction with pressure of reactants is an example of differential equation
ECONOMICS : The equation of Solow - Swan model is an example of differential equation.
PROBLEMS FACED BY STUDENTS IN DIFFERENTIAL EQUATIONS
Differential Equation is one of the most important branches of mathematics, but students face difficulties in dealing with the Differential Equations Assignments :
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