Class 12 Maths - KERALA

Differential Equations

The Differential Equations chapter in Class 12 Kerala SCERT Mathematics bridges calculus with real-world problem-solving. Students learn to find the order and degree of differential equations and solve them using three primary methods: Variable Separable, Homogeneous Differential Equations, and Linear Differential Equations. This chapter is exceptionally scoring in the Kerala Board examinations, frequently testing both procedural skills and word problems involving exponential growth and decay. Mastering this chapter is crucial not only for securing high board marks but also for understanding advanced mathematical modeling in science and engineering.

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Key Concepts

Order and Degree

The order of a differential equation is the highest derivative present, while the degree is the highest power of that highest derivative, provided the equation is polynomial in derivatives.

General and Particular Solutions

A solution containing arbitrary constants is the general solution, whereas a solution free from arbitrary constants obtained by applying initial conditions is a particular solution.

Variable Separable Method

A technique used when all terms containing x can be grouped with dx and all terms containing y with dy, allowing direct integration on both sides.

Homogeneous Differential Equations

Equations where every term is of the same degree, solved by substituting y = vx or x = vy to reduce them to variable separable form.

Linear Differential Equations

Equations of the form dy/dx + Py = Q (or dx/dy + Px = Q) solved by finding an integrating factor (I.F.) equal to e raised to the integral of P with respect to the independent variable.

Important Formulas

dy/dx = f(x)g(y) => (1/g(y))dy = f(x)dx

Homogeneous substitution: y = vx => dy/dx = v + x(dv/dx)

Linear Differential Equation (Type 1): dy/dx + Py = Q, where I.F. = e^(int P dx)

Solution of Linear DE (Type 1): y * (I.F.) = int (Q * I.F.) dx + C

Linear Differential Equation (Type 2): dx/dy + Px = Q, where I.F. = e^(int P dy)

Solution of Linear DE (Type 2): x * (I.F.) = int (Q * I.F.) dy + C

Board Exam Info

In the Kerala (SCERT) Class 12 Mathematics board examination, Differential Equations typically carries around 8 to 12 marks. Common question types include 1-mark questions on finding order and degree, 3-mark questions on solving homogeneous or variable separable equations, and 4 to 6-mark essay questions based on solving Linear Differential Equations or word problems involving curves and growth rates.

Frequently Asked Questions

How do I find the degree of a differential equation if it contains trigonometric functions of derivatives?

If a differential equation cannot be expressed as a polynomial equation in derivatives (like sin(dy/dx) or e^(dy/dx)), its degree is not defined.

How do I know whether to substitute y = vx or x = vy in a homogeneous differential equation?

Check the complexity of the terms. If the expression is easily manageable as a function of y/x, substitute y = vx. If it's easier as x/y, substitute x = vy.

Is the constant of integration 'C' mandatory in the general solution?

Yes, omitting the constant of integration 'C' in a general solution will result in loss of marks in the board exam.

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