Class 12 Mathematics (041) · Chapter 9 NotesDifferential Equations
Master Class 12 Differential Equations with NCERT-based notes. Learn order, degree, variable separable, homogeneous, and linear differential equation methods.
Differential Equations represent a significant leap in mathematical study, moving from static algebraic equations to dynamic relationships involving rates of change. In this chapter, students explore equations that involve derivatives of a dependent variable with respect to one or more independent variables. These equations are fundamental in scientific modeling, appearing frequently in fields such as Physics, Biology, and Economics to describe phenomena like population growth, radioactive decay, and planetary motion. The curriculum focuses on ordinary differential equations, specifically those involving a single independent variable. Students will learn to classify these equations by their order and degree, distinguish between general and particular solutions, and master three primary techniques for solving first-order, first-degree equations: the variable separable method, homogeneous equations, and linear differential equations. Understanding these concepts is essential for grasping how calculus is applied to solve real-world problems where the relationship between quantities is expressed through their derivatives.
What you'll learn
1Define and identify ordinary differential equations and their components.
2Determine the order and degree of various differential equations.
3Distinguish between general solutions and particular solutions of an equation.
4Solve first-order equations using the method of separation of variables.
5Identify and solve homogeneous differential equations using appropriate substitutions.
6Apply the integrating factor method to solve first-order linear differential equations.
Chapter at a glance
01Basic Concepts and Definitions
02Formation of Differential Equations
03Solution of Differential Equations
04Differential Equations of First Order and First Degree
05Applications of Differential Equations
Detailed chapter notes
01
Basic Concepts and Classification
A differential equation is an equation that involves the derivative of a dependent variable with respect to an independent variable. While standard algebraic equations like x² - 5x + 6 = 0 have numerical solutions, differential equations have functions as their solutions. The chapter focuses on Ordinary Differential Equations (ODE), which involve derivatives with respect to only one independent variable. Two fundamental properties used to classify these equations are order and degree.
OrderThe order of a differential equation is the order of the highest-order derivative present in the equation.
DegreeThe degree is the power of the highest-order derivative, provided the equation is a polynomial in its derivatives.
A degree is not defined if the equation cannot be expressed as a polynomial in derivatives (e.g., if a derivative is inside a sine or exponential function).
02
General and Particular Solutions
Solving a differential equation means finding a function that satisfies the relationship defined by the derivatives. There are two main types of solutions. A general solution (or primitive) is one that contains arbitrary constants. The number of these constants always matches the order of the differential equation. A particular solution is derived from the general solution by assigning specific values to the arbitrary constants based on given initial conditions or boundary values.
General SolutionContains 'n' arbitrary constants for an nth-order equation.
Particular SolutionContains zero arbitrary constants.
Solution CurveThe geometric representation of a solution to a differential equation.
03
Method of Variable Separable
This technique is applicable when a first-order, first-degree differential equation can be rearranged so that all terms involving the dependent variable 'y' are on one side with 'dy', and all terms involving the independent variable 'x' are on the other side with 'dx'. Once the variables are separated into the form h(y) dy = g(x) dx, both sides are integrated to find the general solution. This is often the most straightforward method for solving simple differential relationships.
Formdy/dx = g(x)h(y).
ProcedureRewrite as (1/h(y)) dy = g(x) dx and integrate.
Constant of integration 'C' must be added to represent the family of curves.
04
Homogeneous Differential Equations
A differential equation is considered homogeneous if it can be expressed in the form dy/dx = F(x, y), where F is a homogeneous function of degree zero. This means that replacing x with λx and y with λy results in the same function. To solve such equations, we use the substitution y = vx (or x = vy). This substitution transforms the homogeneous equation into a variable separable form involving 'v' and 'x', which can then be integrated and solved.
Homogeneous FunctionF(λx, λy) = λ⁰ F(x, y).
SubstitutionLet y = vx, then dy/dx = v + x(dv/dx).
Final StepAfter integrating, replace 'v' with y/x to get the solution in terms of original variables.
05
Linear Differential Equations
A first-order linear differential equation typically takes the form dy/dx + Py = Q, where P and Q are either constants or functions of x only. To solve this, we calculate an Integrating Factor (I.F.), which is defined as e raised to the power of the integral of P with respect to x. Multiplying the entire equation by this I.F. makes the left side a perfect derivative of (y × I.F.). Integrating both sides then yields the general solution.
Standard Formdy/dx + Py = Q.
Integrating Factor (I.F.): e^∫P dx.
General Solution Formulay × (I.F.) = ∫(Q × I.F.) dx + C.
Alternative Formdx/dy + P₁x = Q₁, where I.F. = e^∫P₁ dy.
Want the complete chapter resources?Topic notes, quizzes and flashcards for Differential Equations.
What is the order of the differential equation d³y/dx³ + 2(d²y/dx²) - 5dy/dx + y = 0?
A1
B2
C3
D4
Show answer
Answer: (C) 3
The order of a differential equation is determined by the highest derivative present. Here, d³y/dx³ is the highest derivative, so the order is 3.
Question 02
A differential equation is formed by eliminating the arbitrary constant from the equation y = ax + b. What is the order of the resulting differential equation?
A0
B1
C2
D3
Show answer
Answer: (B) 1
Differentiating y = ax + b once gives dy/dx = a, which is a first-order differential equation. The order equals the number of times we differentiate to eliminate all arbitrary constants.
Question 03
The order of the differential equation d²y/dx² + (dy/dx)³ = 0 is:
A1
B2
C3
D4
Show answer
Answer: (B) 2
The order of a differential equation is determined by the highest derivative present. Here, the highest derivative is d²y/dx², which is the second derivative, so the order is 2.
Question 04
The order of the differential equation d²y/dx² + (dy/dx)³ = x is:
A1
B2
C3
D4
Show answer
Answer: (B) 2
Order is determined by the highest derivative present. Here, the highest derivative is d²y/dx², which is of order 2.
Question 05
A differential equation of the form dy/dx = f(x) is used to model the rate of change of a quantity. Which of the following represents the general solution after integration?
Ay = ∫f(x)dx + C
By = f(x) + C
Cdy = f(x)
Dy = [f(x)]²
Show answer
Answer: (A) y = ∫f(x)dx + C
The general solution of dy/dx = f(x) is obtained by integrating both sides with respect to x, giving y = ∫f(x)dx + C, where C is the constant of integration.
Ready for more practice?Unlock the full quiz for this chapter.
What is the difference between order and degree of a differential equation?
The order is the highest derivative present in the equation (e.g., second derivative means order 2). The degree is the power of that highest derivative, but only if the equation is a polynomial in its derivatives. Order is always defined, whereas degree may not be defined if derivatives are part of non-polynomial functions like sin(y') or e^y'.
How many arbitrary constants are in a particular solution of a 4th order differential equation?
A particular solution contains zero arbitrary constants. While a general solution of a 4th order differential equation would contain four arbitrary constants, the particular solution is found by assigning specific values to those constants based on given conditions.
When should I use the substitution y = vx?
You should use the substitution y = vx when you identify the differential equation as being homogeneous. This occurs when the function f(x, y) in dy/dx = f(x, y) is a homogeneous function of degree zero, meaning f(λx, λy) = f(x, y).
What is an Integrating Factor and why is it used?
An Integrating Factor (I.F.) is a function used to solve linear differential equations. For the form dy/dx + Py = Q, the I.F. is e^∫P dx. It is used because multiplying the linear equation by this factor transforms the left-hand side into the derivative of the product (y × I.F.), making it easy to integrate.
Can the degree of a differential equation be a fraction?
No. By definition, the degree of a differential equation is the highest power (positive integral index) of the highest order derivative involved. It must be a positive integer. If the equation involves radicals or fractions in the derivatives, it must be rationalized first.