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2016
Ordinary Differential Equations Test-01
Covered topics: Formulation of differential
equations; Equations of first order and first
degree, integrating factor; Orthogonal
trajectory; Equations of first order but not
of first degree, Clairaut's equation, singular
solution.
Ramanasri
S H O P N O - 4 2 , 1 S T F L O O R , N E A R R A P I D F L O U R M I L L S , O L D R A J E N D E R N A G A R , N E W D E L H I C O N T A C T : 8 7 5 0 7 0 6 2 6 2 / 6 3 6 3 / 6 4 6 4
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 2
1. Solve 23 xdy
xy y edx
(10 Marks)
![Page 3: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/3.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 3
2. Show that the equations(4 3 1) (3 2 1) 0x y dx x y dy represents a family of
hyperbolas having as asymptotes the lines 0;x y 2 1 0x y (15 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 4
3. Solve 23 4y xp p (10 Marks)
![Page 5: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/5.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 5
4. Solve2 2
2 2
1xdx ydy x y
ydx xdy x y
(10 Marks)
![Page 6: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/6.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 6
5. Solve 22y x ap ap . Find singular solution and interpret it geometrically (15 Marks)
![Page 7: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/7.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 7
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 8
6. Solve2
2
(log )log
dy y y yy
dx x x (15 Marks)
![Page 9: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/9.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 9
7. Find the general solutions of 2 (2 ) 0 ( 0)ayp x b p y a (15 Marks)
![Page 10: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/10.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 10
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 11
8. Solve 3 23dy
x y x ydx
(10 Marks)
![Page 12: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/12.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 12
9. Find the value of constant , for following differential equation is an exact
2 2(2 3 ) (3 )y ydyxe y x e
dx . Further, this , solve the differential equations. (15 Marks)
![Page 13: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/13.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 13
10. Solve4
6
dy x y
dx x y
(10 Marks)
![Page 14: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/14.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 14
11. Show that orthogonal trajectory of a system of con-focal ellipse is self orthogonal(10 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 15
12. Solve log xdyx y y xyedx
(10Marks)
![Page 16: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/16.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 16
13. Solve the differential equation 2 2 2( )( ) ( 1)px y px y P wheredy
dx,by reducing it to
Clairaut’s from using suitable substitutions (15 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 17
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 18
14. Solve1
cos sin22
dyy x x
dx (10 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 19
15. Solve2 2 2 2( 2 ) ( ) 0y xy x y dx x xy x y dy (10 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 20
16. Reduce the equation( )( ) 2 ;px y py x p wheredy
pdx
to Clairaut’s equation and hence
solve it (15 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 21
17. Find the orthogonal trajectory of a system of co-axial circle 2 2 2 0x y gx c , where is g
parameter (10 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 22
18. Solve 2 2 2 2 1dy
xy x y x ydx
(10 Marks)
![Page 23: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/23.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 23
19. 2 2 2 2( )(1 ) 2( )(1 )( ) ( ) 0;x y p x y p x yp x yp wheredy
pdx
, by reducing it to
Clairaut's form by using suitable substitution (15 Marks)
![Page 24: Ordinary Differential Equations Test-01mathematicsoptional.com/wp-content/uploads/2016/06/ODE... · 2016 Ordinary Differential Equations Test-01 Covered topics: Formulation of differential](https://reader030.vdocuments.site/reader030/viewer/2022040108/5f07cce37e708231d41ecf72/html5/thumbnails/24.jpg)
Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 24
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 25
20. Solve12 tan(1 ) ( ) 0y dyy x edx
(15 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 26
21. Solve the equation 2 2 2(2 ) 0x p yp x y y using the substitution y u andxy v and find
its singular solution, wheredy
pdx
(15 Marks)
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 27
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Ramanasri Institute
Reputed Institute for IAS, IFoS Exams Page 28
22. Solve 32 2( ) 0x
dyxy e x y
dx
(15 Marks)