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Class XII Mathematics

Chapter 8: Differential Equations

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

Chapter Overview

Welcome to Class XII Mathematics: Differential Equations. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

IntegralsApplication of Derivatives

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Definition, order, and degree

Concept Explanation

A differential equation is an equation involving an unknown function and its derivatives. Order is the highest derivative present. Degree is the power of the highest order derivative, provided the equation is a polynomial in derivatives.

Mathematical Representation
\text{Order} = n \, (\text{highest } d^ny/dx^n), \quad \text{Degree} = k \, (\text{power of } d^ny/dx^n)
Study Guideline: To find the degree, ensure the equation is free of fractional powers and radicals in derivatives, and that derivatives are not inside transcendental functions like sin or log.

2General and particular solutions of differential equations

Concept Explanation

The general solution of an n-th order differential equation contains n arbitrary constants. A particular solution is obtained by assigning specific values to these constants using initial or boundary conditions.

Mathematical Representation
y = c_1 e^x + c_2 e^{-x} \, (\text{General}), \quad y = 2e^x - e^{-x} \, (\text{Particular})
Study Guideline: The number of arbitrary constants in the general solution matches the order of the differential equation; the particular solution has zero arbitrary constants.

3Separable variable differential equations

Concept Explanation

Separable differential equations can be solved by grouping all terms in 'y' on one side with dy, and all terms in 'x' on the other side with dx, and integrating both sides independently.

Mathematical Representation
\frac{dy}{dx} = g(x)h(y) \implies \int \frac{1}{h(y)} \, dy = \int g(x) \, dx
Study Guideline: Move the differentials (dx and dy) to the numerators on opposite sides before integrating.

4Homogeneous differential equations

Concept Explanation

A differential equation is homogeneous if it can be written as dy/dx = F(y/x). It is solved by substituting y = vx, which transforms it into a separable equation in v and x.

Mathematical Representation
\frac{dy}{dx} = f(x,y) \quad \text{where } f(tx, ty) = f(x,y) \implies \text{Substitute } y = vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}
Study Guideline: After replacing dy/dx with v + x(dv/dx), separate the variables v and x, integrate, and substitute v = y/x at the end.

5Linear differential equations first order

Concept Explanation

A first-order linear differential equation is of the form dy/dx + Py = Q, where P and Q are functions of x. It is solved by multiplying the equation by an Integrating Factor (IF) to make it integrable.

Mathematical Representation
\frac{dy}{dx} + P y = Q \implies \text{IF} = e^{\int P \, dx} \implies y \cdot \text{IF} = \int (Q \cdot \text{IF}) \, dx + C
Study Guideline: Ensure the coefficient of dy/dx is exactly 1 before identifying P and Q. Integrate P first to find the integrating factor.