# NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.2

Scientific disciplines including Physics, Chemistry, Biology, Mathematics, etc., have been subjects of interest to the human race since the beginning of time. Among the prominent scientific disciplines, Mathematics particularly has a long-drawn history which can be traced back to many early civilizations. The reason is that Mathematics holds great importance in the ordering and smooth functioning of human activities as well as of a majority of human-made or human-controlled operations. Mathematics is thus a critical academic discipline that is taught to students from the beginning of their education.It is also composed of a variety of sub-disciplines, which are vital knowledge systems in themselves. Additionally, Mathematics also has an immeasurable potential for research and experimentation, which makes it ideal for young learners. Within the NCERT academic curriculum, students who wish to pursue Mathematics in greater depth opt for the Mathematics Stream in Classes 11 and 12. Therefore, it goes without saying that Mathematics is an inseparable part of the prescribed NCERT academic curriculum for the students of Class 12. The academic curriculum prescribed by NCERT for Class 12 is aimed at laying a strong foundation in the fundamental sub-disciplines of Mathematics to prepare students for their future endeavours.

This implies the importance of easy access to reliable learning resources like the NCERT Solutions as trustworthy reference materials to facilitate self-learning. The prescribed NCERT academic syllabus for Mathematics in Class 12 is made up of thirteen extensively researched chapters. These chapters have been organised into an ordered sequence for ease of learning. Chapter 9 is titled Differential Equations. Ex 9.2 Class 12 is the second comprehensive assessment, which is a part of this chapter. It has to be objectively comprehended and thoroughly practised to succeed in the examination.

The Extramarks learning platform has compiled the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 to facilitate the learning process of students. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been compiled in collaboration with renowned subject experts to ensure great quality content. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 could be effectively utilised as an efficient resource for self-study and revision by students. The nuances of the latest updated NCERT academic syllabus for Mathematics have been carefully considered in the preparation of the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2.

**NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.2) Exercise 9.2**

Widespread apprehension and indecisiveness concerning the theme of Differential Equations have been observed among students for a long time. This attitude has been a major cause of concern for students and their instructors alike. Students often find the subject of Differential Equations challenging and as a consequence, may lose interest in it. Differential Equations can be quite an interesting theme if one has the necessary conceptual clarity to complement their efforts to understand the subject. It is also clear that Differential Equations and Exercise 9.2 for Class 12 are critical components of the mathematics curriculum determined for Class 12.As a result, the NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.2 can be useful in easing students’ concerns.. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been designed by Extramarks keeping the general concerns and queries of students and teachers in view. The easy-to-understand format of the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 can prove crucial in aiding and encouraging self-learning among students.

Differential Equations are a basic and vital part of the wider academic arena of Mathematics as a subject of research. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 provide clear-cut and simple solutions to Exercise 9.2 of the chapter on Differential Equations. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been compiled with careful consideration of the methods of classroom teaching. Therefore, the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 do not exhibit major discrepancies between classroom lectures and the compiled solutions.

Extramarks is committed to providing dependable, genuine, and descriptive NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 to motivate students to engage with the themes of Differential Equations by moving a step forward from their previous apprehensions. Ex 9.2 Class 12 is a well-organized and logically-structured format of the calculations provided as a part of the NCERT Solutions Class 12 Maths Chapter 9 Exercise 9.2 would help students adequately abide by the ideal format for problem-solving and would clarify their doubts step-by-step.

**Important Topics Covered in NCERT Solution Class 12 Chapter 9 (Exercise 9.2) **

Chapter 9 titled Differential Equations, covers a range of important topics, such as comprehensive questions asking students to verify whether the given function is a solution to the corresponding differential equation. Types of functions, namely implicit and explicit functions have also been covered. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 cover all the essential topics which are a part of Exercise 9.2 of Chapter 9. The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 provide detailed, step-by-step solutions to the problems which comprise Exercise 9.2 Class 12. The formulae used to do the calculations have also been mentioned and covered in detail in order to ensure that students have total clarity about the logical sequence of the calculations. Extramarks is providing the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 to complement the hard work and perseverance of students.

**Access NCERT Solutions for Class 12 Maths Chapter 9 – Permutations and Combinations**

**NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Exercise 9.2**

Differential Equations have often been perceived and upheld as a dynamic, complex and versatile academic theme within Mathematics. However, within the academic syllabus for Mathematics in Class 12 prescribed by the NCERT, the theme of Differential Equations occupies a prominent place. Differential Equations is indeed a theme comprising conceptually-rich content. This is why it is also regarded as an opening into a wider expanse of academic research in the field of Mathematics. Therefore, it is imperative for students who are looking forward to pursuing a career in Mathematics, to improve their conceptual clarity and comprehension of the theme of Differential Equations.

It is highly recommended that students continue with the regular and consistent practise of questions that are part of the theme of Differential Equations. This would be of great assistance in enhancing the retention abilities of students, which is essential to acquiring and retaining important formulas and their respective derivatives. In accordance with this, students must review and practise the exercises that are part of their textbook with the aid of quality reference material like the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2. Regular practise of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.2 will help students achieve excellence in the topic of differential equations.

Through constant practise of the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2, students can become adept at appropriately comprehending the problems presented to them. Accordingly, they would be able to construct an adequate response that caters to the demands of particular questions on the topic of Differential Equations. Students would be able to easily choose the right approach to the practical application of the formulae they have acquired and retained in order to solve questions.

The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been engineered in the form of dynamically consumable learning resources to aid students during the preparation for their exams. Therefore, the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been compiled very selectively through the inclusion of the most efficient ways of solving long-drawn problems related to the theme of Differential Equations.

Extramarks provides high-quality, authentic, and comprehensively explained NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2. These solutions are available through the Extramarks learning platform, are curated by reputed subject experts, and can act as a versatile learning resource for the students.

The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been compiled through immense efforts by Extramarks. The Extramarks learning platform has been cautiously working in order to ensure that the updates of the revised NCERT academic curriculum find a place in the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2. These solutions have been organised into an easy-to-understand and user-friendly framework to aid the self-learning of students. This has been done in a comprehensive manner for the convenience of students so that they can appropriately follow the logical sequence of calculations. It is vital for students to gain knowledge about and retain important formulae and their derivatives. It is also equally essential for them to be proficient in the practical application of this knowledge studied miscellaneously, in order to solve problems. This is a skill which can be achieved through continuous practise and revision. Therefore, Extramarks provides the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 as comprehensive and curated reference materials with quality content.

Extramarks is a reliable, reputed, and efficient learning platform. Extramarks provides high-quality, comprehensive, and easy-to-understand academic content like the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 to complement the efforts of students. Additionally, the Extramarks has other well-structured and well-researched content like the NCERT Solutions Class 1, NCERT Solutions Class 2, NCERT Solutions Class 3, NCERT Solutions Class 4, NCERT Solutions Class 5, NCERT Solutions Class 6, NCERT Solutions Class 7, NCERT Solutions Class 8, NCERT Solutions Class 9, NCERT Solutions Class 10, NCERT Solutions Class 11 and NCERT Solutions Class 12.

**Benefits of NCERT Solutions Class 12 Maths Chapter 9 Exercise 9.2 **

**Benefits of NCERT Solutions Class 12 Maths Chapter 9 Exercise 9.2 **

Differential Equations is a complex topic that contains a variety of information that must be dealt with in various ways and through various learning methods.While at the micro level, it is imperative for students to be familiar with important formulae and their respective derivatives and preferably have them in their memory, conceptual clarity at the holistic level is also mandatory. It is a difficult task to accommodate all of the students’ questions and doubts during classroom teaching sessions and to adequately respond to all of them.These situations may lead to the accumulation of unresolved doubts and queries, which may become a hindrance to attaining the necessary conceptual clarity. This deficiency may pose a significant challenge to students’ sincere efforts to perform well in their exams.

Therefore, the Extramarks learning platform is providing crucial learning resources like the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 to complement the guidance of teachers as well as the sincere efforts of students. The well-organised structure of the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 prepared by Extramarks can be utilised by students to clarify their doubts by themselves through self-learning and regular practice.

The NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 have been carefully crafted to ensure that students are able to appropriately follow the logical sequence of steps in long-drawn calculations. It is imperative for students to not only retain crucial formulas but also be able to practically apply these formulas to solving problems. This skill can be honed through consistent practise and focused revision methods.Therefore, Extramarks provides the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2 as an easily accessible learning resource with quality content for the convenience of reference whenever required. Instead of searching everywhere for answers to complex Calculus problems, students can conveniently access practise material from the Extramarks website. The Extramarks website is committed to providing quality, versatile academic content, all in one place.

**Q.1 **

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ {\text{y = e}}^{\text{x}}\text{+ 1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}\text{:}\text{}\text{}\text{y\u201d \u2013 y\u2019 = 0}\end{array}

**Ans**

\begin{array}{l}{\text{y = e}}^{\text{x}}\text{+ 1}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ {\text{y\u2019 = e}}^{\text{x}}\text{}\mathrm{\dots}\left(\text{i}\right)\\ \text{Differentiating equation}\left(\text{i}\right)\text{w}\text{.r}\text{.t}\text{. x, we get}\\ {\text{y\u201d = e}}^{\text{x}}\\ \text{Substituting values of y\u2019 and y\u201d in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation y\u201d \u2013 y\u2019 = 0, we get}\\ {\text{y\u201d \u2013 y\u2019 =e}}^{\text{x}}{\text{\u2013 e}}^{\text{x}}\text{= 0 = R}\text{.H}\text{.S}\text{.}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.2**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ {\text{y = x}}^{\text{2}}\text{+2x + C}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{:}\text{}\text{}\text{y\u2019 \u20132x \u20132 = 0}\end{array}

**Ans**

\begin{array}{l}{\text{y = x}}^{\text{2}}\text{+2x + C}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{y\u2019 = 2x + 2}\\ \text{Substituting values of y\u2019 in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation y\u2019 \u20132x \u20132 = 0, we get}\\ \text{y\u2019 \u2013 2x \u20132 = 2x + 2 \u20132x \u20132}\\ \text{}\text{}\text{= 0 = R}\text{.H}\text{.S}\text{.}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.3**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ \text{y = cos x + C}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{:}\text{}\text{}\text{y\u2019 + sinx = 0}\end{array}

**Ans**

\begin{array}{l}\text{y = cos x + C}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{y\u2019 = \u2013sinx}\\ \text{Substituting values of y\u2019 in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation y\u2019 + sinx = 0, we get}\\ \text{y\u2019 + sinx = \u2013sinx + sinx = 0 = R}\text{.H}\text{.S}\text{.}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.4**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ \text{y =}\sqrt{{\text{1 + x}}^{\text{2}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{:}\text{}\text{}\text{y\u2019 =}\frac{\text{xy}}{{\text{1 + x}}^{\text{2}}}\end{array}

**Ans**

\begin{array}{l}\text{y =}\sqrt{{\text{1 + x}}^{\text{2}}}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{y\u2019 =}\frac{\text{x}}{\sqrt{{\text{1 + x}}^{\text{2}}}}\\ \text{Substituting values of y in R}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation y\u2019 =}\frac{\text{xy}}{{\text{1 + x}}^{\text{2}}}\text{, we get}\\ \text{R}\text{.H}\text{.S}\text{. =}\frac{\text{xy}}{{\text{1 + x}}^{\text{2}}}\\ \text{}\text{\hspace{0.17em}}\text{=}\frac{\text{x}\left(\sqrt{{\text{1 + x}}^{\text{2}}}\right)}{{\text{1 + x}}^{\text{2}}}\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{=}\frac{\text{x}}{\sqrt{{\text{1 + x}}^{\text{2}}}}\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{= y\u2019 = L}\text{.H}\text{.S}\text{.}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.5**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ \text{y = Ax}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{:}\text{}\text{}\text{xy\u2019 = y}\text{}\text{}\left(\text{x}\ne \text{0}\right)\end{array}

**Ans**

\begin{array}{l}\text{y = Ax}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{y\u2019 = A}\\ \text{Substituting values of y\u2019 in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation xy\u2019 = y, we get}\\ \text{L}\text{.H}\text{.S}\text{. = xy\u2019}\\ \text{}\text{= x}\left(\text{A}\right)\\ \text{}\text{= Ax = y = R}\text{.H}\text{.S}\text{.}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.6**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ \text{y = x sinx}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{: xy\u2019 = y + x}\sqrt{{\text{x}}^{\text{2}}{\text{\u2013 y}}^{\text{2}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{x}\ne \text{0}\text{\hspace{0.17em}}\text{and x > y or x < \u2013y}\right)\end{array}

**Ans**

\begin{array}{l}\text{y = xsin x}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{y\u2019 = xcos x + sinx}\\ \text{Substituting values of y\u2019 in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation xy\u2019 = y + x}\sqrt{{\text{x}}^{\text{2}}{\text{\u2013 y}}^{\text{2}}}\text{, we get}\\ \text{L}\text{.H}\text{.S}\text{. = xy\u2019 = x}\left(\text{xcosx + sinx}\right)\\ {\text{= x}}^{\text{2}}\text{cosx + xsinx}\\ \text{R}\text{.H}\text{.S}\text{. = xsinx + x}\sqrt{{\text{x}}^{\text{2}}\text{\u2013}{\left(\text{xsinx}\right)}^{\text{2}}}\\ {\text{= xsinx + x}}^{\text{2}}\sqrt{{\text{1 \u2013 sin}}^{\text{2}}\text{x}}\\ {\text{= xsinx + x}}^{\text{2}}\text{cosx}\\ \text{So, L}\text{.H}\text{.S}\text{. = R}\text{.H}\text{.S}\text{.}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.7**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ \text{xy = logy + C : y\u2019 =}\frac{{\text{y}}^{\text{2}}}{\text{1 \u2013 xy}}\left(\text{xy}\ne \text{0}\right)\end{array}

**Ans**

\begin{array}{l}\text{xy = logy + C}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{xy\u2019 + y =}\frac{\text{1}}{\text{y}}\text{y\u2019 + 0}\\ {\text{xyy\u2019 + y}}^{\text{2}}\text{= y\u2019}\\ {\text{y}}^{\text{2}}\text{= y\u2019 \u2013 xyy\u2019}\\ \text{= y\u2019}\left(\text{1 \u2013 xy}\right)\\ \text{y\u2019 =}\frac{{\text{y}}^{\text{2}}}{\left(\text{1 \u2013 xy}\right)}\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.8**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ \text{y \u2013 cosy = x :}\left(\text{ysiny + cosy + x}\right)\text{y\u2019 = y}\end{array}

**Ans**

\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y-\mathrm{cos}y=x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\dots}\left(i\right)\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{y\u2019}+\mathrm{sin}y.y\u2018=1\\ y\u2018\left(1+\mathrm{sin}y\right)=1\\ \text{}\text{}y\u2018=\frac{1}{1+\mathrm{sin}y}\\ \text{Substituting values of y\u2019 in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation}\left(\text{ysiny + cosy + x}\right)\text{y\u2019 = y, we get}\\ \left(y\mathrm{sin}y+\mathrm{cos}y+x\right)y\u2018=\left(y\mathrm{sin}y+\mathrm{cos}y+x\right)\frac{1}{\left(1+\mathrm{sin}y\right)}\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(y\mathrm{sin}y+\mathrm{cos}y+y-\mathrm{cos}y\right)\frac{1}{\left(1+\mathrm{sin}y\right)}\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\left[\text{From equation}\left(\text{i}\right)\right]\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(y\mathrm{sin}y+y\right)\frac{1}{\left(1+\mathrm{sin}y\right)}\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=y\left(\mathrm{sin}y+1\right)\frac{1}{\left(1+\mathrm{sin}y\right)}\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=y=R.H.S.\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.9**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ x+y=ta{n}^{\u20131}y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}:\text{}{y}^{2}y\u2018+{y}^{2}+1=0\end{array}

**Ans**

\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}x+y={\mathrm{tan}}^{\u20131}y\text{\hspace{0.17em}}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{1+y\u2019}=\left(\frac{1}{1+{y}^{2}}\right)y\u2018\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{1}+\text{\hspace{0.17em}}\text{y\u2019}\right)\left(1+{y}^{2}\right)=y\u2018\\ 1+{y}^{2}+y\u2018+y\u2018{y}^{2}=y\u2018\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1+{y}^{2}+y\u2018{y}^{2}=0\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.10**

\begin{array}{l}\text{Verify that the given functions}\left(\text{explicit or implicit}\right)\text{is a solution of the corresponding differential equation:}\\ y=\sqrt{{a}^{2}-{x}^{2}}\text{\hspace{0.17em}}x\in \left(-a,a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}:\text{}x+y\frac{dy}{dx}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(y\ne 0\right)\end{array}

**Ans**

\begin{array}{l}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=\sqrt{{a}^{2}-{x}^{2}}\text{\hspace{0.17em}}\\ \text{Differentiating w}\text{.r}\text{.t}\text{. x, we get}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{dy}{dx}=\frac{-x}{\sqrt{{a}^{2}-{x}^{2}}}\\ \text{}\text{}\text{}=\frac{-x}{y}\\ \text{Substituting the value of}\frac{dy}{dx}\text{in L}\text{.H}\text{.S}\text{. of given differential}\\ \text{equation}x+y\frac{dy}{dx}=0,\text{we get}\\ \text{L}\text{.H}\text{.S}\text{.}=x+y\frac{dy}{dx}\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x+y\left(\frac{-x}{y}\right)\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x-x\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0=R.H.S.\\ \text{Thus,the given function is the solution of the corresponding}\\ \text{differential equation}\text{.}\end{array}

**Q.11 **The number of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0 (B) 2 (C) 3 (D) 4

**Ans**

Since, number of constants in a differential equation of order n is equal to its order i.e., n.

Thus, number of arbitrary constants in a differential equation of fourth order are 4.

Thus, option D is correct.

**Q.12** The number of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3 (B) 2 (C) 1 (D) 0

**Ans**

In a particular solution, there are no arbitrary constants.

∴ Number of arbitrary constants = 0

Thus, option D is correct.

##### FAQs (Frequently Asked Questions)

## 1. Que 1.Where to find reliable NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2?

The Extramarks website provides authentic and high-quality online versions of the **NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2**. The content available on the Extramarks learning portal is curated by renowned subject experts and is reliable, up-to-date, and comprehensive. The content under the **NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2** has been compiled in consideration of the latest NCERT academic curriculum. The solutions provided have been engineered in order to ensure efficiency, simplicity, and effectiveness. It has also been ensured that there would be a minimum discrepancies between the content being delivered to students by the teachers in the classroom and the online learning resources being provided to them by Extramarks.

## 2. What are the benefits of the NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2?

The **NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2** curated by the Extramarks website are comprehensive, easy to understand, and well-organised. These solutions will complement the hard work and sincere efforts of students and will aid and encourage self-learning among them. Students seeking answers to unresolved questions and doubts will find these solutions useful and organised according to the latest NCERT syllabus. The Extramarks learning platform provides the **NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2** assuring high-quality and easy-to-understand calculations.

## 3. Is Differential Equations a complex and difficult topic in the NCERT prescribed curriculum for Class 12 for Mathematics?

While Differential Equations can indeed be considered a versatile and complex theme which covers a variety of topics within the prescribed NCERT academic curriculum for Mathematics in Class 12, they are quite useful. Not only are they fundamental areas of study for a wide variety of research fields and professional arenas of study, but they are also quite interesting once one has the necessary conceptual clarity. The **NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.2** made available by Extramarks can be very helpful in cultivating this clarity.