Differential Equations Applications: Types and Applications

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Differential equations are significantly applied in academics as well as in real life. Various disciplines such as pure and applied mathematics, physics, and engineering are concerned with the properties of differential equations of various types. However, differential equations used to solve real-life problems might not necessarily be directly solvable.

Many fundamental laws of physics and chemistry are often formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. There are a number of named differential equations used in various fields, such as the partial differentiation equation, the wave equation, the heat equation, and the Black-Scholes equation. Here, we will discuss various applications of differential equations in mathematics as well as in real life.

Table of Content

What is a differential equation?
Types of differential equation
Equation Order
Applications of differential equation
Things to Remember
Sample Questions

Key Terms: Differential equation, variable, functions, derivatives, Rate of change, complex systems, wave equation, heat equation, Black-Scholes equation

What is a Differential Equation?

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A differential equation, in mathematics, can be referred to as an equation that relates to one or more functions and their derivatives. The said Functions usually represent physical quantities, and their derivatives represent their rates of change. The differential equation creates a relationship between these two.

differential equation

Differential Equation

The study of differential equations is concerned mainly with the study of their solutions, i.e. the set of functions satisfying each equation, and the properties of their solutions. However, the explicit formulae can be used to solve only the simplest equations. Most of the properties of solutions of a given equation are often determined without computing them.

Read More: Applications of Derivatives

Types of Differential Equations

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Differential equations can be categorized into three types:

Ordinary differential equations
Partial differential equations
Non-linear differential equations

Ordinary differential equations:

An equation that contains an unknown function of one real or complex variable x, its derivatives along with some given functions of x, is called an ordinary differential equation. The unknown function is generally represented by a variable (often denoted as y) that depends on x. Therefore, x is referred to as the independent variable of the equation.

Partial differential equations:

A partial differential equation is one that has unknown multivariable functions and their partial derivatives and is named in contrast to ordinary differential equations. The function is often considered to be ‘unknown’ to be solved for. It is similar to how x is considered to be an unknown number to be solved for in an algebraic equation like

x² − 3x + 2 = 0.

Non-linear differential equations:

A non-linear differential equation is a system of differential equations that is not linear in the unknown function and its derivatives. There are hardly any known methods of solving these equations, and therefore the methods typically depend upon the equation having particular symmetries. This is because non-linear differential equations can exhibit very complicated behavior over extended time intervals.

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Equation Order

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Differential equations are described by their order, determined by the term with the highest derivatives. They are classified into:

First-order differential equation – equation containing only first derivatives
Second-order differential equation – equation containing the second derivative

and so on.

Read More: Methods of Integration

Applications of Differential Equations

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Ordinary differential equations are utilized in the real world to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum and to elucidate thermodynamics concepts. Moreover, they are used in the medical field to check the growth of diseases in graphical representation.

Partial differential equations can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These physical phenomena which seem to be distinct can actually be formalized in terms of PDEs. While ordinary differential equations model one-dimensional dynamical systems, partial differential equations model multi-dimensional systems.

Application of Differential Equations

Application of Differential Equations

Pure mathematics focuses on the existence and uniqueness of solutions, while applied math emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play a crucial role in modeling virtually every physical, technical, or biological process- be it celestial motion, bridge design, or interactions between neurons.

The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found applications. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations.

Things to Remember

Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Often when a closed-form expression for the solutions of a differential equation is not available, solutions may be approximated numerically using computers.
The theory of dynamic systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Differential equations first came into existence with the invention of Calculus by Newton and Leibniz. In his work “Methodus fluxionium et Serierum Infinitarum”, Newton listed three types of differential equations.
Differential equations that describe natural phenomena almost always have only first and second-order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth-order partial differential equation.

Sample Questions

Ques: Can differential equations be applied in real life? [4 marks]

Ans. Yes, differential equations are applied in many disciplines ranging from medical, chemical engineering to economics. Since the differential equations have an exceptional capability of foreseeing the world around us, they’re applied to explain an array of disciplines, a number of which are mentioned below:

Exponential growth and decomposition
Growth of population across different species over time
Modification in return on investment over time
Money flow/circulation or optimum investment strategies
Modeling the cancer growth or the spread of a disease
Demonstration of motion of electricity, the motion of waves, the motion of a spring or pendulum system.

Ques: Explain the solution of differential equations. [4 marks]

Ans. The overall solution of the differential equation is the correlation between the variables x and y which is received after removing the derivatives (i.e. integration) where the relation includes arbitrary constants to represent the order of an equation. The answer of the first-order equation includes one arbitrary whereas the second- order differential equation includes two arbitrary constants. If specific values are given to arbitrary constants, the overall solution of the differential equation is received. So as to resolve the first-order differential equation of first degree, some standard forms are here to get the overall solution as follows:

Linear equation
Homogeneous equation
Non-homogeneous differential equation
Exact differential equation
Variable desperate method

Ques: Find the general solution of the following differential equation: [2 marks]
dt/dx = (1 + x²) ( 1+ t²)

Ans. The given differential equation is dt/dx = (1 + x²) ( 1+ t²)

dt( 1+ t²) = (1 + x²)dx

By integrating each side of the above equation, we get

\(\int\)dt/( 1+ t²) = \(\int\)(1 + x²)dx

tan^-1 t = \(\int\)dx \(\int\)dx²

tan-1 t = x + x³/x + C

The above equation is the required general solution of the differential equation.

Ques: Find the general solution of the differential equation given below [2 marks]
dt/dz = e^{z + t}

Ans. We have,

dt/dz = e^{z + t}

Using the law of exponent, we get

dt/dz = e^z + e^t

By separating variables by variable separable procedure, we get

e^-t dt = e^z dz

Now taking integration of both the side, we get

\(\int\)e^-t dt = \(\int\)e^z dz

On integrating, we get

-e^-t = e^z + C
e^z + e^-t = - C Or e^z + e^-t = c

Ques: Find the particular solution of a differential equation which satisfies the below condition: [3 marks]
dy/dx = 3x² – 4 ; y(0) = 4

Ans. We’ll first find the general solution of a differential equation. To do this, we’ll integrate each side to find y

dy/dx = 3x² – 1

y = \(\int\)(3x² – 1) dx

y = x³ –x + 4

This is our general solution, to find the particular solution of a differential equation, we’ll apply the initial condition given to us ( y= 4 and x = 0) and solve for C:

y = x³ – x + c

Now, we apply our initial conditions (x = 0, y = 4) and solve for C, which can give us our particular solution:

4 = (0)³ – 0 + C

Now, we will solve for C

4 = C

y = x³ –x + 4

Hence, the particular solution of a differential equation is x³ –x + 4

Ques: Find the particular solution of a differential equation which satisfies the below condition: [3 marks]
dy/dx = 1/x² ; y(1) = 4

Ans. We’ll first find the general solution of a differential equation. To do this, we will integrate each side to find y

dy/dx = 1/x² ;

y(1) = 4

This is our general solution, to find the particular solution of a differential equation, we will apply the initial condition given to us ( y= 4 and x = 1) and solve for C:

y = -1/x+ c

y= \(\int\)(1/x²) dx

y = x^-1 / -1 + C

Now, we apply our initial conditions (x = 1, y = 4) and solve for C, which can give us our particular solution:

4 = -1/1+C

Now, we’ll solve for C

4 = -1 + C

5 = C

y = -1/x + 5

Hence, the particular solution of a differential equation is y = -1/x + 5

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Differential Equations Applications: Types and Applications

What is a Differential Equation?

Types of Differential Equations

Equation Order

Applications of Differential Equations

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
By using the properties of definite integrals, evaluate the integral: \(∫_0^π log(1+cosx)dx\)

2.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

3.
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

4.
Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)

5.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

6.
Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

Similar Mathematics Concepts

Comments

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Differential Equations Applications: Types and Applications

What is a Differential Equation?

Types of Differential Equations

Equation Order

Applications of Differential Equations

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. By using the properties of definite integrals, evaluate the integral: \(∫_0^π log(1+cosx)dx\)

2. Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

3. Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

4. Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)

5. If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that (i) \((A+B)'=A'+B' \)(ii) \((A-B)'=A'-B'\)

6. Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
By using the properties of definite integrals, evaluate the integral: \(∫_0^π log(1+cosx)dx\)

2.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

3.
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

4.
Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)

5.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

6.
Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).