Master Differential Equations – Class 12 Maths Chapter 9 | Concepts, NCERT Solutions & PYQs

Here are some important points about differential equations:

• Definition of a Differential Equation: A differential equation is an equation that involves the derivative (or derivatives) of a dependent variable with respect to an independent variable (or variables). These equations are fundamental in various fields, including Physics, Chemistry, Biology, Anthropology, Geology, and Economics, highlighting their prime importance in modern scientific investigations.

  • For example, the equation dy/dx = g(x), where y = f(x), is a differential equation. Similarly, dy/dx + x/y = 0 is a differential equation because it involves variables and the derivative of y with respect to x.

• Types of Differential Equations:

  • Ordinary Differential Equation: This type involves derivatives of the dependent variable with respect to only one independent variable. The sources primarily focus on ordinary differential equations. An example is d²y/dx² + (dy/dx)² = 0.
  • Partial Differential Equations: These involve derivatives with respect to more than one independent variable, but the provided text confines its study to ordinary differential equations.

• Basic Concepts:

  • Order of a Differential Equation: This is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable present in the given differential equation. For instance, dy/dx = e^x has an order of one, d²y/dx² + y = 0 has an order of two, and d³y/dx³ + (d²y/dx²)² + x = 0 has an order of three.
  • Degree of a Differential Equation: To determine the degree, the differential equation must be a polynomial equation in its derivatives (e.g., y', y'', y'''). When it is a polynomial equation in derivatives, the degree is the highest power (positive integral index) of the highest order derivative involved.
    • If an equation is not a polynomial in its derivatives, its degree is not defined. For example, sin(dy/dx) + dy/dx = 0 is not a polynomial equation in y' and its degree cannot be defined.
    • Both order and degree (if defined) are always positive integers.

• Solutions of a Differential Equation:

  • Unlike algebraic equations whose solutions are numbers, the solution of a differential equation is a function that satisfies the given equation. The curve represented by y = φ(x) is known as the solution curve or integral curve.
  • General Solution (Primitive): This solution contains arbitrary constants (parameters), and the number of arbitrary constants is equal to the order of the differential equation. For example, y = a sin(x + b) is a general solution to d²y/dx² + y = 0 because it contains two arbitrary constants, a and b.
  • Particular Solution: This is a solution derived from the general solution by assigning particular values to the arbitrary constants, meaning it is free from arbitrary constants. For instance, y = 2sin(x + π/4) is a particular solution of d²y/dx² + y = 0.

• Methods of Solving First Order, First Degree Differential Equations: The sources outline three primary methods:

  • Differential Equations with Variables Separable: In this method, a first-order, first-degree differential equation dy/dx = F(x, y) can be solved if F(x, y) can be expressed as a product of a function of x and a function of y, i.e., g(x)h(y). The variables are then separated to (1/h(y)) dy = g(x) dx, allowing for integration of both sides to find the solution.
  • Homogeneous Differential Equations: A function F(x, y) is homogeneous of degree n if F(λx, λy) = λ^n F(x, y) for any non-zero constant λ. A differential equation dy/dx = F(x, y) is homogeneous if F(x, y) is a homogeneous function of degree zero. To solve such equations, a substitution y = vx (or x = vy if dx/dy form) is made, which transforms the equation into a variables separable form.
  • Linear Differential Equations: A first-order linear differential equation has the form dy/dx + Py = Q, where P and Q are constants or functions of x only. Alternatively, it can be dx/dy + P₁x = Q₁, where P₁ and Q₁ are constants or functions of y only. The solution involves finding an Integrating Factor (I.F.), which for the dy/dx form is e^(∫Pdx). The general solution is then given by y (I.F) = ∫(Q * I.F) dx + C. Similarly, for the dx/dy form, I.F = e^(∫P₁dy), and the solution is x (I.F) = ∫(Q₁ * I.F) dy + C.

• Historical Context of Differential Equations: The birth of differential equations is attributed to Gottfried Wilhelm Freiherr Leibnitz on November 11, 1675, when he introduced the symbols ∫ and dy while trying to find a curve with prescribed tangents. Leibnitz also discovered the 'method of separation of variables' in 1691 and formulated the 'method of solving homogeneous differential equations of the first order' and the 'method of solving a linear differential equation of the first-order' shortly after. The term 'integral' for the solution was coined by James Bernoulli in 1690, while the word 'solution' was first used by Joseph Louis Lagrange in 1774 and advocated by Jules Henri Poincare. John Bernoulli is credited with naming the 'method of separation of variables' and highlighting the intricate nature of differential equations.

 

 Chapter 9: Differential Equations (Class 12 CBSE Maths – 2025–26):

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🧠 A. Key Concepts in Differential Equations

1. Definition

A differential equation is an equation that involves a function and its derivatives.

2. Order and Degree

• Order: The order of the highest derivative.

• Degree: Power of the highest order derivative (after removing roots/fractions).

3. General & Particular Solutions

• General Solution includes arbitrary constant(s).

• Particular Solution is found by applying given conditions (initial value problems).

4. Methods of Solving

a) Variable Separation Method

$$\frac{dy}{dx} = f(x)g(y) \Rightarrow \frac{1}{g(y)}dy = f(x)dx$$

Integrate both sides to find the solution.

b) Homogeneous Equations

If $\frac{dy}{dx} = f(x, y)$ and $f(x, y)$ is homogeneous (same degree), use substitution:

$$y = vx \Rightarrow \frac{dy}{dx} = v + x\frac{dv}{dx}$$

c) Linear Differential Equation

$$\frac{dy}{dx} + Py = Q \Rightarrow \text{I.F.} = e^{\int Pdx}, \quad y \cdot \text{I.F.} = \int Q \cdot \text{I.F.} dx$$

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📘 B. NCERT Exercise-Wise Detailed Solutions

Exercise 9.1 – Form differential equations

From general functions like $y = c_1e^x + c_2e^{-x}$, eliminate constants by differentiation.

Exercise 9.2 – Variable separable method

Solve:

$$\frac{dy}{dx} = xy \Rightarrow \frac{1}{y}dy = xdx \Rightarrow \ln y = \frac{x^2}{2} + C \Rightarrow y = Ce^{x^2/2}$$

Exercise 9.3 – Homogeneous equations

Use substitution $y = vx$.
Example:

$$\frac{dy}{dx} = \frac{x+y}{x} \Rightarrow \frac{dy}{dx} = 1 + \frac{y}{x} = 1 + v \Rightarrow \frac{dv}{dx} = \frac{1 + v - v}{x}$$

Exercise 9.4 – Linear equations

$\frac{dy}{dx} + Py = Q$.
Example:

$$\frac{dy}{dx} - \frac{2}{x}y = x^2 \Rightarrow \text{I.F.} = x^{-2} \Rightarrow yx^{-2} = \int x^0 dx = x + C \Rightarrow y = x^3 + Cx^2$$

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🏆 C. CBSE Previous Year Questions (PYQs)

✅ 2023 (3 marks)

Solve:

$$\frac{dy}{dx} = \frac{2x + 3y + 1}{2x + 3y + 2}$$

Hint: Put $z = 2x + 3y + 2$, reduce to separable form.

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✅ 2022 (5 marks)

Find particular solution of

$$\frac{dy}{dx} = x + y, \quad y(0) = 1$$

Solution:
Linear form ⇒ $\frac{dy}{dx} - y = x$
I.F. = $e^{-x}$

$$y \cdot e^{-x} = \int x \cdot e^{-x} dx = -xe^{-x} - e^{-x} + C \Rightarrow y = -x - 1 + Ce^x$$

Apply $y(0) = 1 \Rightarrow C = 2$
Final Answer: $y = -x - 1 + 2e^x$

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✅ 2020 (4 marks)

Solve:

$$\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$$

Solution: Homogeneous form. Use substitution $y = vx$, solve for $v$.

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#Class12Maths, #DifferentialEquations, #LinearEquations, #VariableSeparation, #HomogeneousEquations, #CBSEPYQs, #NCERTSolutions, #Chapter9Maths