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.
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 represent 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 Equation
For a better understanding of differential equations. Let us understand it with the help of an example :
Let us consider a simple equation :
x2+ 2x+ 1 = 0
It has a solution x=-1
But in the 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 derivatives 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
This solution is called the general solution.
Some of the major classifications of differential equations are :
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 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' +a2= 0 is non-linear because a2 is 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 because a' is not multiplied by a constant
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' +t2 a= 0 is homogeneous
a' +t2 a=t+t2 is non-homogeneous
When you know about the behavior of the model under different circumstances is known as an analytic solution. It also referred to as a closed-form solution.
Differential Equation is very important in the biological, physical, and technical process (bridge design, celestial motion, and interaction between neurons). Differential equations have open-form solutions.
The applications of differential equations in real life are; chemistry, biology, physics, and the other areas of natural sciences and economics, and engineering.
Few Examples of its Application are:
Chemistry: The rate law in a chemical reaction with the pressure of reactants is an example of a differential equation
Economics: The equation of the Solow - Swan model is an example of the differential equation.
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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