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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

Ramanasri Institute

Reputed Institute for IAS, IFoS Exams Page 2

1. Solve 23 xdy

xy y edx

(10 Marks)

Ramanasri Institute

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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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3. Solve 23 4y xp p (10 Marks)

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4. Solve2 2

2 2

1xdx ydy x y

ydx xdy x y

(10 Marks)

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5. Solve 22y x ap ap . Find singular solution and interpret it geometrically (15 Marks)

Ramanasri Institute

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Ramanasri Institute

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6. Solve2

2

(log )log

dy y y yy

dx x x (15 Marks)

Ramanasri Institute

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7. Find the general solutions of 2 (2 ) 0 ( 0)ayp x b p y a (15 Marks)

Ramanasri Institute

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Ramanasri Institute

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8. Solve 3 23dy

x y x ydx

(10 Marks)

Ramanasri Institute

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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)

Ramanasri Institute

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10. Solve4

6

dy x y

dx x y

(10 Marks)

Ramanasri Institute

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11. Show that orthogonal trajectory of a system of con-focal ellipse is self orthogonal(10 Marks)

Ramanasri Institute

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12. Solve log xdyx y y xyedx

(10Marks)

Ramanasri Institute

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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)

Ramanasri Institute

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Ramanasri Institute

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14. Solve1

cos sin22

dyy x x

dx (10 Marks)

Ramanasri Institute

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15. Solve2 2 2 2( 2 ) ( ) 0y xy x y dx x xy x y dy (10 Marks)

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16. Reduce the equation( )( ) 2 ;px y py x p wheredy

pdx

to Clairaut’s equation and hence

solve it (15 Marks)

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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18. Solve 2 2 2 2 1dy

xy x y x ydx

(10 Marks)

Ramanasri Institute

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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)

Ramanasri Institute

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Ramanasri Institute

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20. Solve12 tan(1 ) ( ) 0y dyy x edx

(15 Marks)

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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)

Ramanasri Institute

Reputed Institute for IAS, IFoS Exams Page 27

Ramanasri Institute

Reputed Institute for IAS, IFoS Exams Page 28

22. Solve 32 2( ) 0x

dyxy e x y

dx

(15 Marks)