CBSE Class 12 Mathematics NCERT Solutions: Differential Equations

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NCERT Solutions PDF Class 12 PDF

This resource provides detailed NCERT Solutions for Class 12 Mathematics, focusing on Chapter 9: Differential Equations. It covers the fundamental concepts of determining the order and degree of differential equations. The solutions offer step-by-step guidance for each problem, clarifying how to identify the highest order derivative and its power to ascertain the order and degree. This chapter is crucial for understanding the behavior and classification of differential equations, which are essential in various fields of science and engineering. These solutions are designed to help students grasp these concepts thoroughly and prepare effectively for their board examinations by offering clear explanations and accurate problem-solving techniques.

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Quick info

BoardCBSE
ClassClass 12
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterDifferential Equations

Chapter summary

This chapter focuses on understanding differential equations, specifically on determining their order and degree. The NCERT Solutions provide clear explanations and step-by-step methods to identify the highest order derivative present in an equation to find its order. It also explains how to determine the degree by looking at the power of the highest order derivative, provided the equation is a polynomial in derivatives. The solutions cover various forms of differential equations, including those where the degree might not be defined.

Learning outcomes

  • Understand the concepts of order and degree of a differential equation.
  • Identify the highest order derivative in a given differential equation.
  • Determine the order of a differential equation.
  • Determine the degree of a differential equation when it is defined.
  • Recognize differential equations where the degree is not defined.

Topics covered

Paper topics

  • Differential Equations
  • Order of a Differential Equation
  • Degree of a Differential Equation
  • Highest Order Derivative
  • Polynomial in Derivatives
  • Undefined Degree

Important topics

  • Determining the Order of a Differential Equation
  • Determining the Degree of a Differential Equation
  • Conditions for Undefined Degree
  • Identifying Highest Order Derivatives

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Questions and Solutions

Question 1

Determine the order and degree (if defined) of the differential equation: \frac{d^4y}{dx^4} + \sin\left(y'''\right) = 0
Solution:

The given differential equation is:

\frac{d^4y}{dx^4} + \sin\left(y'''\right) = 0

We can rewrite the equation using prime notation as:

y^{iv} + \sin(y''') = 0

The order of a differential equation is determined by the highest order derivative present. In this equation, the highest order derivative is \(y^{iv}\) (the fourth derivative of y with respect to x). Therefore, the order of the differential equation is 4.

The degree of a differential equation is the power of the highest order derivative when the equation is expressed as a polynomial in its derivatives. However, this equation contains the term \(\sin(y''')\), which is not a polynomial function of \(y'''\). Therefore, the degree of this differential equation is not defined.

Question 2

Determine the order and degree (if defined) of the differential equation: y' + 5y = 0
Solution:

The given differential equation is:

y' + 5y = 0

The order of the differential equation is determined by the highest order derivative present. In this equation, the highest order derivative is \(y'\) (the first derivative of y with respect to x). Therefore, the order of the differential equation is 1.

The degree of a differential equation is the power of the highest order derivative. The equation is a polynomial in \(y'\), and the power of \(y'\) is 1. Therefore, the degree of the differential equation is 1.

Question 3

Determine the order and degree (if defined) of the differential equation: \left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0
Solution:

The given differential equation is:

\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0

The order of the differential equation is determined by the highest order derivative present. The derivatives in the equation are \(\frac{ds}{dt}\) (first order) and \(\frac{d^2s}{dt^2}\) (second order). The highest order derivative is \(\frac{d^2s}{dt^2}\). Therefore, the order of the differential equation is 2.

The degree of a differential equation is the power of the highest order derivative when the equation is expressed as a polynomial in its derivatives. The equation is a polynomial in \(\frac{ds}{dt}\) and \(\frac{d^2s}{dt^2}\). The highest order derivative is \(\frac{d^2s}{dt^2}\), and its power is 1. Therefore, the degree of the differential equation is 1.

Question 4

Determine the order and degree (if defined) of the differential equation: \left(\frac{d^2y}{dx^2}\right)^2 + \cos\left(\frac{dy}{dx}\right) = 0
Solution:

The given differential equation is:

\left(\frac{d^2y}{dx^2}\right)^2 + \cos\left(\frac{dy}{dx}\right) = 0

The order of the differential equation is determined by the highest order derivative present. The highest order derivative is \(\frac{d^2y}{dx^2}\) (the second derivative of y with respect to x). Therefore, the order of the differential equation is 2.

The degree of a differential equation is the power of the highest order derivative when the equation is expressed as a polynomial in its derivatives. This equation contains the term \(\cos\left(\frac{dy}{dx}\right)\), which is not a polynomial function of \(\frac{dy}{dx}\). Therefore, the degree of this differential equation is not defined.

Question 5

Determine the order and degree (if defined) of the differential equation: \frac{d^2y}{dx^2} = \cos 3x + \sin 3x
Solution:

The given differential equation is:

\frac{d^2y}{dx^2} = \cos 3x + \sin 3x

We can rewrite the equation as:

\frac{d^2y}{dx^2} - \cos 3x - \sin 3x = 0

The order of the differential equation is determined by the highest order derivative present. In this equation, the highest order derivative is \(\frac{d^2y}{dx^2}\) (the second derivative of y with respect to x). Therefore, the order of the differential equation is 2.

The degree of a differential equation is the power of the highest order derivative when the equation is expressed as a polynomial in its derivatives. The equation is a polynomial in \(\frac{d^2y}{dx^2}\), and the power of \(\frac{d^2y}{dx^2}\) is 1. Therefore, the degree of the differential equation is 1.

Common mistakes

  • Confusing order with degree.
  • Incorrectly identifying the highest order derivative.
  • Assuming degree is always defined, especially when trigonometric functions of derivatives are involved.
  • Errors in algebraic manipulation when simplifying equations to find the degree.

Revision tips

  • Focus on identifying the highest order derivative first to determine the order.
  • Check if the differential equation is a polynomial in its derivatives before determining the degree.
  • Pay close attention to terms like sin(y'), cos(y''), etc., as they often lead to an undefined degree.
  • Practice with a variety of examples to build confidence in identifying order and degree.

Practice MCQs

Q1. What is the order of the differential equation \(\frac{d^4y}{dx^4} + \sin(y''') = 0\)?

Q2. What is the degree of the differential equation \(y' + 5y = 0\)?

Q3. For the differential equation \(\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0\), what is its order?

Q4. What is the degree of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^2 + \cos\left(\frac{dy}{dx}\right) = 0\)?

Q5. What is the order of the differential equation \(\frac{d^2y}{dx^2} = \cos 3x + \sin 3x\)?

Frequently asked questions

What is the order of a differential equation?

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

How is the degree of a differential equation determined?

The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in its derivatives. If the equation is not a polynomial in its derivatives, the degree is not defined.

When is the degree of a differential equation not defined?

The degree is not defined if the differential equation involves derivatives within functions like trigonometric, exponential, or logarithmic functions (e.g., \(\sin(y')\), \(e^{y''}\)), or if it cannot be expressed as a polynomial in its derivatives.

Are the questions in this exercise about solving differential equations?

No, these specific questions focus on determining the order and degree of given differential equations, not on solving them to find the general or particular solution.

Why is it important to find the order and degree?

Understanding the order and degree is fundamental to classifying and analyzing differential equations. It helps in choosing appropriate methods for solving them and understanding their behavior.

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