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NCERT Solutions · Class 12 · Chapter 9: Differential Equations

NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1

Exercise 9.1: Order and degree. Order = the highest derivative present. Degree = the power of that highest derivative once the equation is a polynomial in all the derivatives; if a derivative sits inside sin, log, e and so on, the degree is not defined.

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Exercise 9.1 questions and solutions

Exercise 9.1, Question 1

Find the order and degree (if defined).
\(\dfrac{d^4y}{dx^4} + \sin(y''') = 0\)
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  1. Highest derivative \(y^{(4)}\); \(y'''\) appears inside sine, so the equation is not a polynomial in the derivatives.
Answer: Order 4; degree not defined

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Exercise 9.1, Question 3

\(\left(\dfrac{ds}{dt}\right)^4 + 3s\dfrac{d^2s}{dt^2} = 0\)
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  1. Highest derivative \(\dfrac{d^2s}{dt^2}\), to the first power.
Answer: Order 2; degree 1

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Exercise 9.1, Question 4

\(\left(\dfrac{d^2y}{dx^2}\right)^2 + \cos\left(\dfrac{dy}{dx}\right) = 0\)
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  1. \(\dfrac{dy}{dx}\) is inside cosine.
Answer: Order 2; degree not defined

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Exercise 9.1, Question 5

\(\dfrac{d^2y}{dx^2} = \cos 3x + \sin 3x\)
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  1. \(y''\) to the first power; functions of x alone do not affect the degree.
Answer: Order 2; degree 1

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Exercise 9.1, Question 10

\(y'' + 2y' + \sin y = 0\)
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  1. \(\sin y\) involves y, not a derivative, so the equation is still a polynomial in \(y', y''\).
Answer: Order 2; degree 1

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Exercise 9.1, Question 11

The degree of \(\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0\) is: (A) 3 (B) 2 (C) 1 (D) not defined
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  1. \(\sin(y')\) makes it non-polynomial in the derivatives.
Answer: (D)

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Exercise 9.1, Question 12

The order of \(2x^2\dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + y = 0\) is: (A) 2 (B) 1 (C) 0 (D) not defined
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  1. Highest derivative: \(\dfrac{d^2y}{dx^2}\).
Answer: (A)

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Done the NCERT exercises? The board paper asks more

Differential Equations has 40 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.

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Differential Equations in our sample papers: Sample paper 1 (question 28) · Sample paper 2 (questions 15, 28, 37) · Sample paper 3 (questions 12, 29) · Sample paper 4 (questions 15, 28) · Sample paper 5 (questions 12, 34).

Also useful: free MCQs and case studies for Differential Equations · formulas for this chapter (Class 12 formula sheet, free PDF) · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan

Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.