NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Ex 9.2
Page No 385:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7787/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m443ecf95.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7787/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_310e8085.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7787/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1efd3e8a.gif)
Now, differentiating equation (1) with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7787/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1e013820.gif)
Substituting the values of
in the given differential equation, we get the L.H.S. as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7787/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_1f27ec50.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7787/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m348c95f8.gif)
Thus, the given function is the solution of the corresponding differential equation.
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7788/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m44ece45e.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7788/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3424a2a8.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7788/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7f84e0d9.gif)
Substituting the value of
in the given differential equation, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7788/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ae65614.gif)
L.H.S. =
= R.H.S.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7788/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1ecf5291.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7790/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_396abd52.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7790/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_118f79fa.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7790/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_1fe9d551.gif)
Substituting the value of
in the given differential equation, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7790/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ae65614.gif)
L.H.S. =
= R.H.S.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7790/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m5228b70.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7791/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7c66fd00.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7791/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6b4ca84d.gif)
Differentiating both sides of the equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7791/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4c86b28c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7791/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4dd19828.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7794/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_45e8dfd1.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7794/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6a9cdb2d.gif)
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7794/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m65969c45.gif)
Substituting the value of
in the given differential equation, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7794/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ae65614.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7794/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m19aa76fd.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7795/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_635d5d71.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7795/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m23f43a47.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7795/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_157d89da.gif)
Substituting the value of
in the given differential equation, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7795/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ae65614.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7795/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m21a9b8a6.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7797/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_65766775.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7797/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4fd65be2.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7797/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m668cc977.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7797/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4dd19828.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7799/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m54afabca.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7799/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_10cd8c5c.gif)
Differentiating both sides of the equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7799/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_683e721e.gif)
Substituting the value of
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7799/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ae65614.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7799/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m78325704.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7801/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m16a9f39.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7801/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_f7ec03b.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7801/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_78421a53.gif)
Substituting the value of
in the given differential equation, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7801/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ae65614.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7801/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m48321cce.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7803/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7a6a5292.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7803/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_5a615185.gif)
Differentiating both sides of this equation with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7803/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6a3ff1a0.gif)
Substituting the value of
in the given differential equation, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7803/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_51dda3f3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7803/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m794e8ebf.gif)
Hence, the given function is the solution of the corresponding differential equation.
Question 11:
The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:
(A) 0 (B) 2 (C) 3 (D) 4
Answer:
We know that the number of constants in the general solution of a differential equation of order n is equal to its order.
Therefore, the number of constants in the general equation of fourth order differential equation is four.
Hence, the correct answer is D.
Question 12:
The numbers of arbitrary constants in the particular solution of a differential equation of third order are:
(A) 3 (B) 2 (C) 1 (D) 0
Answer:
In a particular solution of a differential equation, there are no arbitrary constants.
Hence, the correct answer is D.