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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1. Solve 23 xdy

xy y edx

(10 Marks)

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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)

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6. Solve2

2

(log )log

dy y y yy

dx x x (15 Marks)

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

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8. Solve 3 23dy

x y x ydx

(10 Marks)

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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)

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10. Solve4

6

dy x y

dx x y

(10 Marks)

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

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12. Solve log xdyx y y xyedx

(10Marks)

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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)

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14. Solve1

cos sin22

dyy x x

dx (10 Marks)

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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)

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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)

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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)

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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)

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22. Solve 32 2( ) 0x

dyxy e x y

dx

(15 Marks)