Differential Equations Notes: Class 12 Mathematics Chapter 9
Class 12 Mathematics Chapter 9 Differential Equations covers the basic terminology of differential equations (order and degree), methods for solving them — variable separable, homogeneous, and linear differential equations — and their use in modeling real-world situations.
Overview of Topics & Subtopics (Click to Jump)
1. Basic Concepts: Order and Degree
Short Note:
• The order of a differential equation is the order of the highest derivative present, while the degree is the power of that highest order derivative, once the equation is free of radicals and fractions involving derivatives.
• A general solution contains arbitrary constants equal in number to the order of the differential equation, while a particular solution is obtained by assigning specific values to these constants using given conditions.
2. Variable Separable and Homogeneous Equations
Short Note:
• In a variable separable differential equation of the form dy/dx = f(x)g(y), the variables can be separated onto opposite sides of the equation and integrated independently.
• A homogeneous differential equation, where dy/dx is a function of y/x alone, is solved using the substitution y = vx, which converts it into a variable separable form in v and x.
3. Linear Differential Equations
Short Note:
• A first order linear differential equation has the standard form dy/dx + Py = Q, where P and Q are functions of x (or constants).
• Such equations are solved by first finding the integrating factor (I.F.) = e^(∫P dx), multiplying throughout by it, and then integrating both sides.
NCERT Textbook Exercise Questions
Define the order and degree of a differential equation with an example.
Answer: The order of a differential equation refers to the order of the highest derivative appearing in it, while the degree refers to the power of that highest order derivative, provided the equation is expressed as a polynomial in its derivatives. For example, in the equation (d²y/dx²)³ + dy/dx = 0, the order is 2 (since the highest derivative is the second derivative) and the degree is 3 (since that derivative is raised to the third power).
Solve the differential equation dy/dx = y.
Answer: Separating the variables gives dy/y = dx, and integrating both sides yields ln|y| = x + C, where C is an arbitrary constant. Simplifying further, this can be rewritten in the exponential form y = Ae^x, where A = e^C is another arbitrary constant, representing exponential growth.
What substitution is used to solve a homogeneous differential equation, and why?
Answer: A homogeneous differential equation is solved by substituting y = vx, where v is treated as a new function of x, so that dy/dx = v + x(dv/dx). This substitution converts the original equation, which is a function of y/x, into a new equation that is separable in terms of v and x, making it possible to integrate and solve.
What is an integrating factor, and how is it used to solve a linear differential equation?
Answer: The integrating factor (I.F.) for a linear differential equation dy/dx + Py = Q is calculated as e^(∫P dx). Multiplying the entire differential equation by this integrating factor transforms the left-hand side into the exact derivative of a product, d/dx(y · I.F.), which can then be integrated directly to obtain the general solution.
