NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Ex 9.5
Page No 406:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7857/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m501d51d2.gif)
Answer:
The given differential equation i.e., (x2 + xy) dy = (x2 + y2) dx can be written as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7857/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_468af196.gif)
This shows that equation (1) is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7857/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_224e91db.gif)
Substituting the values of v and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7857/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7857/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_61b044ec.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7857/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3cb673b1.gif)
This is the required solution of the given differential equation.
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7859/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m973c521.gif)
Answer:
The given differential equation is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7859/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m670d98c.gif)
Thus, the given equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
Differentiating both sides with respect to x, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7859/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_224e91db.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7859/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7859/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3a99fd6d.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7859/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m33b8352d.gif)
This is the required solution of the given differential equation.
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_37f98764.gif)
Answer:
The given differential equation is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_37f98764.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_26b48392.gif)
Thus, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_388a264a.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7861/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_20d6a717.gif)
This is the required solution of the given differential equation.
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_48d97c5a.gif)
Answer:
The given differential equation is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_48d97c5a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1ef08d50.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_52ac7312.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7863/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_5d1f8e64.gif)
This is the required solution of the given differential equation.
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2012_02_21_18_18_00/mathmlequation2229679458588616949.png)
Answer:
The given differential equation is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7865/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_me1b4e05.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7865/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m546757fb.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7865/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m29d2fdbc.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7865/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7865/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_73686725.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7865/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_66457b63.gif)
This is the required solution for the given differential equation.
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7867/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6a7383fe.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Correction.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7867/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of v and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7867/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7867/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m33584c21.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7867/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6201ef90.gif)
This is the required solution of the given differential equation.
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6fd1a0e.gif)
Answer:
The given differential equation is:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m69384e9f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7ea34bd0.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m52d3f8f9.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6c6223.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_1b66b121.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7871/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7c82eb1c.gif)
This is the required solution of the given differential equation.
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7874/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2fbda755.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7874/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m31e95b94.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7874/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7874/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7874/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7061f891.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7874/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_5f897286.gif)
This is the required solution of the given differential equation.
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3e600719.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4a386dc4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_71d2abdd.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6ca14e08.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m50de6cb2.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_mc87225f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m33f40725.gif)
Therefore, equation (1) becomes:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7877/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m212ae962.gif)
This is the required solution of the given differential equation.
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_1684f6df.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m45bf726b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_18adc76b.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
x = vy
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_afe66dd.gif)
Substituting the values of x and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7bd1cf62.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6e6cbffd.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7880/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_35c908e9.gif)
This is the required solution of the given differential equation.
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2012_02_21_18_20_53/mathmlequation6844457755488916695.png)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_65a02c22.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_6d73669a.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7b817b7d.gif)
Now, y = 1 at x = 1.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_d018409.gif)
Substituting the value of 2k in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7881/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_7629563.gif)
This is the required solution of the given differential equation.
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2c45a64.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_379506c9.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2b742fb5.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m4ddd2dd7.gif)
Now, y = 1 at x = 1.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_330f2f01.gif)
Substituting
in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_261def3d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7883/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_351c847a.gif)
This is the required solution of the given differential equation.
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2012_02_21_18_33_42/mathmlequation4401542383907660145.png)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_11c56719.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve this differential equation, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3dff1adc.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m5578fce1.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m7eda5bd.gif)
Now,
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m442f6de2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_mfe44a1f.gif)
Substituting C = e in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7885/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_56a0dfd5.gif)
This is the required solution of the given differential equation.
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_2a64122f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_47f61fb1.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the values of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m1fb6564d.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m749f6ef3.gif)
This is the required solution of the given differential equation.
Now, y = 0 at x = 1.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m2439f1a8.gif)
Substituting C = e in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7887/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_123984f5.gif)
This is the required solution of the given differential equation.
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6d8af437.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_1d36072b.gif)
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m6305e560.gif)
Substituting the value of y and
in equation (1), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_708e8096.gif)
Integrating both sides, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_598c85d8.gif)
Now, y = 2 at x = 1.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_445b41c7.gif)
Substituting C = –1 in equation (2), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7888/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m3904717f.gif)
This is the required solution of the given differential equation.
Question 16:
A homogeneous differential equation of the form
can be solved by making the substitution
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7890/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_397d3c70.gif)
A. y = vx
B. v = yx
C. x = vy
D. x = v
Answer:
For solving the homogeneous equation of the form
, we need to make the substitution as x = vy.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7890/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_397d3c70.gif)
Hence, the correct answer is C.
Page No 407:
Question 17:
Which of the following is a homogeneous differential equation?
A. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4d154b48.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_4d154b48.gif)
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3f584b72.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_3f584b72.gif)
C. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_65846662.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_65846662.gif)
D. ![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2012_02_21_18_36_00/mathmlequation7821902014782140484.png)
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2012_02_21_18_36_00/mathmlequation7821902014782140484.png)
Answer:
Function F(x, y) is said to be the homogenous function of degree n, if
F(λx, λy) = λn F(x, y) for any non-zero constant (λ).
Consider the equation given in alternativeD:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/238/7892/NCERT_12-11-08_Gopal_12_Maths_Ex-9.1_12_MNK_SG_html_m40c4c2ed.gif)
Hence, the differential equation given in alternative D is a homogenous equation.