NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Ex 9.3
Page No 391:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7807/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m45b65fa8.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7807/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m45b65fa8.gif)
Differentiating both sides of the given equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7807/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1a54e347.gif)
Again, differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7807/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_1a87606f.gif)
Hence, the required differential equation of the given curve is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7807/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_24724108.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7807/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_24724108.gif)
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7808/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_2f40307e.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7808/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_2f40307e.gif)
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7808/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7802fdc6.gif)
Again, differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7808/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_215ba94c.gif)
Dividing equation (2) by equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7808/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m43aa41a3.gif)
This is the required differential equation of the given curve.
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_28cc9ab6.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m78cb8a4c.gif)
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m99a1b4f.gif)
Again, differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3fd4c7f2.gif)
Multiplying equation (1) with (2) and then adding it to equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m758236cb.gif)
Now, multiplying equation (1) with 3 and subtracting equation (2) from it, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1b991034.gif)
Substituting the values of
in equation (3), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m47bbbdf8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7809/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m129795fb.gif)
This is the required differential equation of the given curve.
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7810/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1e6b17cc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7810/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7008e2d6.gif)
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7810/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2e6d8a5b.gif)
Multiplying equation (1) with 2 and then subtracting it from equation (2), we get:
y’-2y=e2x2a+2bx+b-e2x2a+2bx⇒y’-2y=be2x …(3)
Differentiating both sides with respect to x, we get:
y”-2y’=2be2x …4Dividing equation (4) by equation (3), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7810/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_mc6d6144.gif)
This is the required differential equation of the given curve.
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7811/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_38cd523b.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7811/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6fe4f635.gif)
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7811/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m388dd07b.gif)
Again, differentiating with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7811/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_31ab3c03.gif)
Adding equations (1) and (3), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7811/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7b23a7f8.gif)
This is the required differential equation of the given curve.
Question 6:
Form the differential equation of the family of circles touching the y-axis at the origin.
Answer:
The centre of the circle touching the y-axis at origin lies on the x-axis.
Let (a, 0) be the centre of the circle.
Since it touches the y-axis at origin, its radius is a.
Now, the equation of the circle with centre (a, 0) and radius (a) is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7812/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m78d856ad.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7812/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m45e68912.jpg)
Differentiating equation (1) with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7812/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3ae6d8d0.gif)
Now, on substituting the value of a in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7812/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_16e6b2c6.gif)
This is the required differential equation.
Question 7:
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Answer:
The equation of the parabola having the vertex at origin and the axis along the positive y-axis is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7813/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_34067d1c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7813/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3482ba8a.jpg)
Differentiating equation (1) with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7813/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7854845e.gif)
Dividing equation (2) by equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7813/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2d1b93ad.gif)
This is the required differential equation.
Question 8:
Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.
Answer:
The equation of the family of ellipses having foci on the y-axis and the centre at origin is as follows:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7814/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_5abb36a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7814/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m4127dc6e.jpg)
Differentiating equation (1) with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7814/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_133b9d3.gif)
Again, differentiating with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7814/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2139adb.gif)
Substituting this value in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7814/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m41b5b7b8.gif)
This is the required differential equation.
Question 9:
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
Answer:
The equation of the family of hyperbolas with the centre at origin and foci along the x-axis is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7815/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_75582719.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7815/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_5a7d2a02.jpg)
Differentiating both sides of equation (1) with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7815/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_d797667.gif)
Again, differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7815/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m22ae9735.gif)
Substituting the value of
in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7815/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6cff76ee.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7815/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m50fb1b86.gif)
This is the required differential equation.
Question 10:
Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
Answer:
Let the centre of the circle on y-axis be (0, b).
The differential equation of the family of circles with centre at (0, b) and radius 3 is as follows:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7816/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_2f7e42bf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7816/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1eae9f72.jpg)
Differentiating equation (1) with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7816/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_2507d7bd.gif)
Substituting the value of (y – b) in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7816/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m691c354f.gif)
This is the required differential equation.
Question 11:
Which of the following differential equations has
as the general solution?
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m61c77c02.gif)
A. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2b67ca89.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2b67ca89.gif)
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3179f792.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3179f792.gif)
C. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_66739e81.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_66739e81.gif)
D. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m39cb6828.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m39cb6828.gif)
Answer:
The given equation is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7f696c14.gif)
Differentiating with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6d5220dd.gif)
Again, differentiating with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7817/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_454aea12.gif)
This is the required differential equation of the given equation of curve.
Hence, the correct answer is B.
Question 12:
Which of the following differential equation has
as one of its particular solution?
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m5960828c.gif)
A. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m5fb757a4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m5fb757a4.gif)
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_574fd33b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_574fd33b.gif)
C. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m4ba12a8e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m4ba12a8e.gif)
D. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m48c7274a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m48c7274a.gif)
Answer:
The given equation of curve is y = x.
Differentiating with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7f7ddac8.gif)
Again, differentiating with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7af9dd40.gif)
Now, on substituting the values of y,
from equation (1) and (2) in each of the given alternatives, we find that only the differential equation given in alternative C is correct.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1b7d7217.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7818/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_ma9cc9cc.gif)
Hence, the correct answer is C.