# NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.1) Exercise 9.1

The most fundamental concepts of Mathematics are used by scientists to determine quantitative solutions to experimental laws. It is applied in multiple fields including Finance, Medicine, Computer Science, Social Sciences, Natural Sciences Engineering, etc. Therefore, it would be correct to state that the concepts of Mathematics are widely applied in the curriculum of multiple subjects in the senior academic years of students.The mathematics curriculum for Class 12 introduces students to a variety of essential mathematical concepts that serve as the foundation for a basic conceptual understanding of a variety of topics at the higher education level.. The students of Class 12 should work hard to build a strong knowledge of mathematical concepts to score well in the board examinations, as well as to be able to apply those concepts in their further studies.

Class 12 Mathematics Chapter 9 is Differential Equations. Students might have had a glimpse of this chapter earlier in Class 11. These equations arise in a variety of applications in Biology, Anthropology, Physics, Geology, Chemistry and much more. Therefore, all modern scientific investigations use the concepts of Differential Equations. In addition, Differential Equations are equations that include one or more functions with their derivatives. There are many ways to solve the problems related to these equations, however, the easiest one is through the explicit use of formulas. Mathematics is a subject that requires rigorous practice. NCERT should be the first reference point for all students. Therefore, Extramarks provides students with the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 so that they can practise, improve their academic performance, and have a successful academic career.

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The Central Board of Secondary Education is an educational board in India for multiple public and private schools, controlled and managed by the Indian Government. It is one of the most significant boards of education in India, as most of the schools in India follow the CBSE board. CBSE conducts the board examinations for Class 10 and Class 12 every year between February to May. It was established in 1929 by a resolution of the Indian government. All the schools affiliated with the CBSE board follow the NCERT curriculum, especially, Classes 10 and 12. The major objectives of NCERT are to promote and coordinate research in areas related to school education, prepare and publish model textbooks, supplementary material, and much more. It also acts as a clearinghouse for ideas and information in matters related to school education, and it acts as a government agency for achieving the goals of universalisation of Elementary Education. The National Council of Educational Research and Training (NCERT) is an independent organisation introduced in 1961 by the Government of India to assist the Central and State Governments with policies and programmes for improvement in school education.

**NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.1) Exercise 9.1**

Class 12 Chapter 9 Differential Equation includes the basic concepts related to Differential Equations, general and particular solutions of a Differential Equation, and the formation of differential equations. Other topics such as some methods to solve a first order – first-degree differential equation and some applications of differential equations in different areas, also form a part of the syllabus for the chapter. This chapter covers a wide range of topics. Extramarks provides NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 to students so that they can easily access authentic solutions without having to search elsewhere.The Extramarks website provides students with multiple learning modules, in-depth performance reports, and much more to clear all their queries and doubts. This helps students focus on their academic goals and succeed in their board examinations.

Differential Equations Exercise 9.1 includes the Order of a Differential Equation and the Degree of a Differential Equation. Students can refer to the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1, for a better understanding of the chapter. Exercise 9.1 Class 12th includes twelve questions in which students have to determine the order and degree of the Differential Equations. In the beginning, students might find this exercise a bit challenging, but the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 provided by Extramarks help the students get a deeper understanding of the concepts of the chapter. This enables students to solve any complicated problem related to the topics of this chapter. Extramarks is a platform that provides students with all the resources they need to score well in any in-school, board, or competitive examination.Along with the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1, Extramarks also provides students with the NCERT Solutions for all the academic sessions and respective subjects. Students can refer to the Extramarks website for NCERT Solutions Class 12, NCERT Solutions Class 11, NCERT Solutions Class 10, and NCERT Solutions Class 9. Other learning materials such as NCERT Solutions Class 8, NCERT Solutions Class 7, NCERT Solutions Class 6, NCERT Solutions Class 5, NCERT Solutions Class 4, NCERT Solutions Class 3, NCERT Solutions Class 2, and NCERT Solutions Class 1 are also available.

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**NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Exercise 9.1**

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**NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Exercise 9.1**

The NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 provided by the Extramarks website are curated by expert teachers at Extramarks. They are properly detailed and incorporated into a step-by-step approach. The NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 walk students through the logic behind each step so that they can have strong and clear concepts. The NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 provided by the Extramarks website are easy to download and can be accessed from any device.Students in Class 12 must have the NCERT curriculum at their fingertips. The NCERT curriculum has multiple provisions that test the students’ basic concepts. Differential Equations, Chapter 7, may be the most difficult chapter of Class 12 Mathematics.NCERT books are written by subject experts, so if students thoroughly go through the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1, they can understand all the concepts of the chapter. These solutions provided by Extramarks make students ready for their board examinations. All the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 are explained step by step with all the details so that the students do not face any conceptual difficulties. Extramarks recommends students thoroughly review these solutions before their board examinations.

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**Q.1 **Determine order and degree (if defined) of differential equation

$\frac{{\mathrm{d}}^{\mathrm{4}}\mathrm{y}}{{\mathrm{dx}}^{\mathrm{4}}}\hspace{0.17em}+\mathrm{\hspace{0.17em}}\mathrm{sin}\left(\mathrm{y}\right)\hspace{0.17em}=\hspace{0.17em}0$

**Ans**

$\begin{array}{l}\frac{{\mathrm{d}}^{\mathrm{4}}\mathrm{y}}{{\mathrm{dx}}^{\mathrm{4}}}+\mathrm{sin}\left(\mathrm{y}\right)=0\Rightarrow \mathrm{y}\mathrm{\hspace{0.17em}}+\mathrm{sin}\left(\mathrm{y}\right)=0\\ \mathrm{The}\mathrm{heighest}\mathrm{order}\mathrm{derivative}\mathrm{present}\mathrm{in}\mathrm{the}\mathrm{differential}\mathrm{equation}\\ \mathrm{is}\frac{{\mathrm{d}}^{\mathrm{4}}\mathrm{y}}{{\mathrm{dx}}^{\mathrm{4}}}.\mathrm{Therefore}\mathrm{its}\mathrm{order}\mathrm{is}4\mathrm{.}\\ \mathrm{The}\mathrm{given}\mathrm{equation}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{polynomial}\mathrm{equation}\mathrm{in}\mathrm{y}\u2018\mathrm{and}\mathrm{degree}\\ \mathrm{of}\mathrm{such}\mathrm{a}\mathrm{differential}\mathrm{equation}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{defined}\mathrm{.}\end{array}$

**Q.2 **

**Ans**

\begin{array}{l}\text{y + 5y=0}\\ \text{The heighest order derivative present in the differential equation}\\ \text{is y}\text{. Therefore its order is 1}\text{.}\\ \text{The given equation is a polynomial equation in y so the heighest}\\ \text{power raised by y is one}\text{. Thus, degree of the differential equation}\\ \text{is 1}\text{.}\end{array}

**Q.3 **

**Ans**

\begin{array}{l}{\left(\frac{\text{ds}}{\text{dt}}\right)}^{\text{4}}\text{+ 3s}\frac{{\text{d}}^{\text{2}}\text{s}}{{\text{dt}}^{\text{2}}}\text{=0}\\ \text{The}\text{\xe2\u20ac\u201e}\text{heighest oder derivative present in differential equation is}\\ \frac{{\text{d}}^{\text{2}}\text{s}}{{\text{dt}}^{\text{2}}}\text{. Therefore, its order is 2}\text{.}\\ \text{Since, given differential equation is a polynomial in}\frac{{\text{d}}^{\text{2}}\text{s}}{{\text{dt}}^{\text{2}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\frac{\text{ds}}{\text{dt}}\text{.}\\ \text{The power raised by}\frac{{\text{d}}^{\text{2}}\text{s}}{{\text{dt}}^{\text{2}}}\text{is 1, so the degree of equation is 1}\text{.}\end{array}

**Q.4 **

**Ans**

\begin{array}{l}{\left(\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\right)}^{\text{2}}\text{+cos}\left(\frac{\text{dy}}{\text{dx}}\right)\text{=0}\\ \text{The}\text{\xe2\u20ac\u201e}\text{heighest oder derivative present in differential equation is}\\ \frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{. Therefore, its order is 2}\text{.}\\ \text{Since, given differential equation is not a polynomial in}\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \text{and}\frac{\text{dy}}{\text{dx}}\text{.}\\ \text{So, its degree is not defined}\text{.}\end{array}

**Q.5 **

**Ans**

\begin{array}{l}\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{=cos3x + sin3x}\Rightarrow \frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}-\text{cos3x}-\text{sin3x=0}\\ \text{The}\text{\xe2\u20ac\u201e}\text{heighest oder derivative present in differential equation is}\\ \frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{. Therefore, its order is 2}\text{.}\\ \text{Since, given differential equation is a polynomial in}\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{\hspace{0.17em}}\text{.}\\ \text{The power raised by}\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{is 1, so the degree of equation is 1}\text{.}\end{array}

**Q.6 **

**Ans**

\begin{array}{l}{\left(\text{y}\right)}^{\text{2}}\text{+}{\left(\text{y}\right)}^{\text{3}}\text{+}{\left(\text{y}\right)}^{\text{4}}\text{+}{\left(\text{y}\right)}^{\text{5}}\text{=0}\\ \text{The heighest order derivative in given differential equation is y}\text{.}\\ \text{So, its order is 3}\text{.}\\ \text{This differential equation is a polynomial in y\u201d\u2019, y\u201d, y\u2019 and y}\text{.}\\ \text{The heighest power raised by y is 2, therefore degree of}\\ \text{differential equation is 2}\text{.}\end{array}

**Q.7 **

\begin{array}{l}\text{Determine order and degree}\left(\text{if defined}\right)\text{of differential equation:}\\ \text{y + 2y}\text{+}\text{y}\text{=}\text{0}\end{array}

**Ans**

$\begin{array}{l}\mathrm{y}\u201d\u2019\; +\; 2\mathrm{y}\u201d+\mathrm{y}\u2018=0\\ \mathrm{The}\mathrm{heighest}\mathrm{order}\mathrm{derivative}\mathrm{in}\mathrm{given}\mathrm{differential}\mathrm{equation}\mathrm{is}\mathrm{y}\u201d\u2019\mathrm{.}\\ \mathrm{So},\mathrm{its}\mathrm{order}\mathrm{is}3\mathrm{.}\\ \mathrm{This}\mathrm{differential}\mathrm{equation}\mathrm{is}\mathrm{a}\mathrm{polynomial}\mathrm{in}\mathrm{y}\u201d\u2019,\mathrm{y}\u201d\mathrm{and}\mathrm{y}\u2018\mathrm{.}\\ \mathrm{The}\mathrm{heighest}\mathrm{power}\mathrm{raised}\mathrm{by}\mathrm{y}\u201d\u2019\mathrm{is}1,\mathrm{therefore}\mathrm{degree}\mathrm{of}\\ \mathrm{differential}\mathrm{equation}\mathrm{is}1\mathrm{.}\end{array}$

**Q.8 **

**Ans**

\begin{array}{l}{\text{y\u2019+ y=e}}^{\text{x}}{\text{\xdey\u2019+ y-e}}^{\text{x}}\text{=0}\\ \text{The heighest order derivative in given differential equation is y\u2019}\text{.}\\ \text{So, its order is 1}\text{.}\\ \text{This differential equation is a polynomial in y\u2019 and y}\text{.}\\ \text{The heighest power raised by y\u2019 is 1, therefore degree of}\\ \text{differential equation is 1}\text{.}\end{array}

**Q.9 **

**Ans**

\begin{array}{l}\text{y+}{\left(\text{y}\right)}^{\text{2}}\text{+2y=0}\\ \text{The heighest order derivative in given differential equation isy}\text{.}\\ \text{So, its order is 2}\text{.}\\ \text{This differential equation is a polynomial in y, y and y}\text{.}\\ \text{The heighest power raised by y is 1, therefore degree of}\\ \text{differential equation is 1}\text{.}\end{array}

**Q.10 **

**Ans**

\begin{array}{l}\text{y+ 2y+ sin y=0}\\ \text{The heighest order derivative in given differential equation is}\text{\hspace{0.17em}}\text{y}\text{.}\\ \text{So, its order is 2}\text{.}\\ \text{This differential equation is a polynomial in y\u201d and y\u2019}\text{.}\\ \text{The heighest power raised by y\u201d is 1, therefore degree of}\\ \text{differential equation is 1}\text{.}\end{array}

**Q.11 **

\begin{array}{l}\text{The degree of the differential equation}\\ {\left(\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\right)}^{\text{3}}\text{+}{\left(\frac{\text{dy}}{\text{dx}}\right)}^{\text{2}}\text{+ sin}\left(\frac{\text{dy}}{\text{dx}}\right)\text{+1}\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{is}\\ \left(\text{A}\right)\text{3}\text{}\text{}\left(\text{B}\right)\text{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\left(\text{C}\right)\text{1}\text{}\text{}\left(\text{D}\right)\text{\hspace{0.17em}}\text{not}\text{\xe2\u20ac\u2039}\text{defined}\end{array}

**Ans**

\begin{array}{l}{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{3}+{\left(\frac{dy}{dx}\right)}^{2}+\mathrm{sin}\left(\frac{dy}{dx}\right)+1=0\\ \text{S}\text{i}\text{n}\text{c}\text{e}\text{given equation is not a polynomial equation in y\u2019}\text{. Therefore,}\\ \text{its degree is not defined}\text{.}\\ \text{Hence, the correct answer is D}\text{.}\end{array}

**Q.12 **

\begin{array}{l}\text{The order of the differential equation}\\ {\text{2x}}^{\text{2}}\frac{{\text{d}}^{\text{2}}\text{y}}{{\text{dx}}^{\text{2}}}\text{\hspace{0.17em}}\text{\u2013}\text{\hspace{0.17em}}\text{3}\frac{\text{dy}}{\text{dx}}\text{\hspace{0.17em}}\text{+}\text{\hspace{0.17em}}\text{y}\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{is}\\ \left(\text{A}\right)\text{3}\text{}\text{}\left(\text{B}\right)\text{2}\text{}\text{}\left(\text{C}\right)\text{0}\text{}\text{}\left(\text{D}\right)\text{not}\text{\xe2\u20ac\u2039}\text{defined}\end{array}

**Ans**

\begin{array}{l}2{x}^{2}\frac{{d}^{2}y}{d{x}^{2}}-3\frac{dy}{dx}+y=0\text{\hspace{0.17em}}\\ \text{T}\text{h}\text{e}\text{highest derivative of given differential equation is}\frac{{d}^{2}y}{d{x}^{2}}.\\ \text{T}\text{h}\text{e}\text{r}\text{e}\text{f}\text{o}\text{r}\text{e},\text{the order of given differential equation is 2}\text{.}\\ \text{Thus, correct option is B}\text{.}\end{array}

##### FAQs (Frequently Asked Questions)

## 1. Que 1.Are the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 available on the Extramarks website?

Yes, the **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 **are available on the Extramarks website. These solutions are curated by the subject experts atExtramarks and can be downloaded on any device very easily as they are available in PDF format. Along with **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 **Extramarks also provides students with various learning modules that help students with an enjoyable and systematic learning process. Extramarks is a platform that has proved that e-learning is very beneficial for students. Students can subscribe to the Extramarks website for easy access to the **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1.**

## 2. Is Class 12 Chapter 9 Differential Equations difficult?

Chapter 9 Differential Equations is not complicated for the students. If students review the **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 **thoroughly, they will be able to review the concepts of the chapter easily and apply those concepts to any complicated problem that can appear in the examinations.

## 3. Is it essential to practise all the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1?

Yes, students should practice all the **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 **as every question helps them in revising all necessary topics in the chapter. Also, practising them increases the speed and accuracy of the students and helps them avoid small calculation mistakes so that they do not lose marks in the board examinations.

## 4. How can students clarify their doubts about the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1?

The **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 **provided by the Extramarks website are incorporated in a step-by-step approach and are properly detailed so that students can understand such concepts easily. If students face any further queries, they can subscribe to the Extramarks website. Learning modules like live doubt-solving sessions, practise tests, and much more, help students clarify their doubts and also track their academic progress.

## 5. Will the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 help students in appearing for other competitive examinations?

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## 6. How can students prepare for the Class 12 Mathematics board examination?

The primary step that students should take in preparation for their Class 12 Mathematics board examination is to go through the **NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1. **Then students should revise the extra questions and revision notes of the chapter. Thereafter, they should thoroughly review the sample papers and the past years’ papers on the curriculum of the subject. This way, students will practice the necessary concepts of the subject, and they will be able to solve any complicated problem that they can encounter during their board examinations.