What are Differential Equations?
When an equation has one or more functions as well as its derivatives, it is known as a differential equation. There are certain terms that we need to take care of while studying differential equations. Those terms are given below:
2. Independent Variable: The dependent variable is dependent on another variable, which is known as the independent variable.
Note: The differential equations may or may not have either one or more than one dependent or independent variable.
Differential Equations have many use cases. They are taken into consideration almost in every field, be it chemistry, mathematics, biology, physics, engineering, and so on. From species of any living organism to rough engineering, chemical decomposition, population, and other areas of research, differential equations play a massive role.
The dependent variables, with consideration of independent variables, when forming a function derivative, then this phenomenon is known as differential equations.
Order and degree are the main terms that ought to be perceived while tackling the differential conditions. Order of a differential condition is the most noteworthy capacity to which the subordinates are brought up in the given condition. Be that as it may, degree then again is the force of the greatest subordinate. For instance, consider the differential condition referenced underneath.
(y’)2 + y’’’ - 2 (y’’)4 = 7y
The equation comprises the third derivative of 'y' as y''' which is the most noteworthy derivative. The power of y''' is 1. Thus, the level of the equation is '1'. Be that as it may, the subsequent derivative is y'' which is raised to power 4 which is the most elevated power of the derivative. Along these lines, the request for the given differential equation is 4.
The Different Types of Differential Equations
There are many different types of differential equations, starting with the basis of the type of variables, the types are:
Types of Differential Equations Based on the Order of Equations:
1. First order of differential equation: When 1 is the highest power of the formed derivatives.
2. Second-order of the differential equation: When 2 is the highest power of the formed derivatives.
3. N(th) order of differential equation: When 'N' is the highest power of the formed derivatives.
Types of differential equations based on homogeneity:
Homogenous differential equations
Non-homogeneous differential equations
Solution of Differential Equations:
Solving a differential equation means finding an equation that does not contain any derivatives. However, this equation should satisfy the differential equation that is being solved. Solving differential equations involves two or more integrations. To determine an appropriate method to solve the differential equation, it is very important to identify the type of differential equation that is being solved. Both general and particular solutions of differential equations can be obtained by using appropriate steps to solve the equation.
Fun facts:
A differential equation will generally have an infinite number of solutions.
A general formula can be derived for the solution of a few differential equations.
FAQs on Differential Equation And Its Types
1. What is a differential equation and how is it identified in mathematics?
A differential equation is an equation that involves one or more functions and their derivatives with respect to one or more independent variables. It is identified by the presence of derivatives (like dy/dx or d2y/dx2) in the equation, which relate a dependent variable to its rate of change.
2. What are the main types of differential equations covered in the CBSE Class 12 Maths syllabus?
Differential equations can be classified based on different criteria:
- Order: First order, second order, or nth order, depending on the highest derivative present.
- Nature: Ordinary Differential Equations (ODEs) involve a single independent variable, while Partial Differential Equations (PDEs) involve more than one.
- Linearity: Linear and Nonlinear differential equations.
- Homogeneity: Homogeneous and Non-homogeneous differential equations.
3. How are order and degree of a differential equation defined and determined?
The order of a differential equation is the highest order of the derivative present in the equation, while the degree is the exponent of the highest order derivative (after removing fractional and negative powers of derivatives, if any). For example, in (d2y/dx2)3 + dy/dx = 0, order is 2, degree is 3.
4. Why is it important to distinguish between linear and nonlinear differential equations when solving problems?
Linear differential equations have solutions that can be found using standard methods, and their principles of superposition apply, making them easier to analyze. Nonlinear equations often require more complex techniques or numerical methods, and their solutions may exhibit unpredictable or chaotic behavior. Correctly identifying the type guides the approach to solving.
5. What real-life phenomena can be modeled using differential equations?
Differential equations are widely used to model real-world scenarios such as:
- Population growth and decay
- Chemical reaction rates
- Spread of diseases in biology
- Motion of projectiles or waves in physics
- Electrical circuits in engineering
- Investment growth in economics
6. How does the general solution of a differential equation differ from a particular solution?
The general solution contains arbitrary constants and represents a family of all possible solutions of the differential equation. A particular solution is obtained by assigning specific values to these constants, often to satisfy given initial or boundary conditions.
7. What steps should be followed to solve a differential equation as per the NCERT/CBSE 2025–26 guidelines?
To solve a differential equation, follow these steps:
- Identify the type and order of the equation
- Simplify if necessary
- Select the appropriate solution method (e.g., variable separable, integrating factor, homogeneous equations)
- Integrate as per the chosen method
- Apply initial or boundary conditions if provided to find particular solutions
8. How can you determine whether a differential equation is homogeneous or non-homogeneous?
A homogeneous differential equation has all terms involving the dependent variable and its derivatives (and possibly multiplied by functions of the independent variable), with zero on the other side. A non-homogeneous equation has a non-zero function or constant on the right side that is not a function of the dependent variable or its derivatives.
9. What are some common misconceptions students have about solving first order differential equations?
Common misconceptions include:
- Confusing the methods for linear vs separable equations
- Forgetting to include the constant of integration
- Incorrectly identifying the independent and dependent variables
- Not checking that the solution satisfies the original equation
10. If a differential equation models the rate of temperature change in a body, what type is it and how is it typically solved?
This is usually a first order linear differential equation representing Newton's Law of Cooling. It is typically solved using the integrating factor method to find how temperature changes over time.
