Maharashtra Board Class 12 Maths Part 2 Chapter 6 Differential Equations PDF Download

6. Differential Equations

Let Us Study

Differential Equation

Order and degree of differential equation

Formation of differential equation

Solution of differential equation

Types of differential equation

Application of differential equation

Let Us Recall

The differentiation and integration of functions and the properties of differentiation and integration

Let Us Learn

6.1.1 Introduction

In physics, chemistry and other sciences we often have to build mathematical models which involves differential equations. We need to find functions which satisfy those differential equations.

6.1.2 Differential Equation

Equation which contains the derivative of a function is called a differential equation. The following are differential equations:

(i) \(\frac{dy}{dx} = \cos x\)

(ii) \(\frac{d^2y}{dx^2} + k^2y = 0\)

(iii) \(\left(\frac{d^2w}{dx^2}\right) - x^2 \frac{dw}{dx} + w = 0\)

(iv) \(\frac{d^2y}{dt^2} + \frac{d^2x}{dt^2} = x\) where x and y are functions of t

(v) \(\frac{d^3y}{dx^3} + x\frac{dy}{dx} - 4xy = 0\) where x is a function of y

(vi) \(r\frac{dr}{d\theta} + \cos \theta = 5\)

Order and Degree of the Differential Equation

The order of a differential equation is the highest order of the derivative appearing in the equation.

The degree of differential equation is the power of the highest ordered derivative present in the equation. To find the degree of the differential equation, we need to have a positive integer as the index of each derivative.

Teacher's Note

A differential equation shows how something changes. Like when we see a plant grow, the growth rate changes every day. This is a real-life example of a differential equation.

Exam Trick

Order = Highest derivative's number (like first, second, third). Degree = Power of that highest derivative. Remember: Order comes first, then degree.

Points to Remember

Order is the highest number of times we differentiate.

Degree is how many times the highest derivative is multiplied.

Order and degree are always whole positive numbers.

We cannot find degree if the equation has sine or cosine of a derivative.

A differential equation with all the same term degrees is homogeneous.

Teacher's Note

When we find order and degree, we first look at all the derivatives in the equation. The highest derivative number gives us the order, and the power of that derivative gives us the degree.

Exam Trick

Order = Count (first derivative = 1, second = 2). Degree = Power (squared = 2, cubed = 3). Simple pattern to remember!

Points to Remember

Find the highest derivative first.

Look at the power of that highest derivative.

Order and degree must always be positive whole numbers.

If you see a square root or other function with a derivative inside, simplify first.

The degree is not defined if a derivative is inside a trigonometric function.

Teacher's Note

Understanding order and degree is like understanding the strength of change. A first-order equation shows simple change, while a second-order shows how the change itself changes. In India, we use these ideas when designing bridges and buildings.

Exam Trick

Remember: Order = how many times you take derivative. Degree = the power of the highest derivative. Write them down as: Order, Degree in your answer.

Points to Remember

Order is always seen in the derivative: dy/dx (order 1), d²y/dx² (order 2).

Degree is the power: (dy/dx)² means degree 2.

Both order and degree are positive integers always.

Write the differential equation in simplest form before finding order and degree.

If the equation has a root like √(d²y/dx²), remove the root first by squaring.

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