Ordinary Differential Equations (ODE)

Transcription

1 Ordinary Differential Equations (ODE) Previous year Questions from 07 to 99 Ramanasri Institute W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : /

2 07. Find the differential equation representing the entire circle in the y- plane. [0 Marks]. Solve the following simultaneous liner differential equations: ( D ) y z e and ( D ) z y e where y and z are functions of independent variable and D d. [8 Marks] 3. If the growth rate of the population of bacteria at time t is proportional to the amount present at the time and population doubles in one week, then how much bacteria s can be epected after 4 weeks? [8 Marks] 4. Consider the differential equation yp ( y ) p y 0 where p substituting u and dv v y reduce the equation to Clairaut s form in terms of uvand, p' hence or otherwise solve du the equation. [0 Marks] 5. Solve the following initial value differential equations 0 y" 4 y' y 0, y(0) 3., y'(0) 0. [7 Marks] sin( ) [9 Marks] d y 6. Solve the differential equation: y 7. Solve that following differential equation using method of variation of parameters y [8 Marks] d y 8. Solve the following initial value problem using Laplace transform: 8sin if 0 where r ( ) 0 if. 06 d y y r y y 9 ( ), (0) 0, '(0) 4 [7 Marks] 9. d y / 3 Find a particular integral of y e sin [0 marks] 0. Show that the family of parabolas y 4c 4c is self orthogonal. [0 marks]. Solve{ y( tan ) cos } 0 [0 marks]. Using the method of variation of parameter solve the differential equation d ( D D ) y e log( ), D [5 marks] 3 3. Find the general solution of the equation d y 4 d y [5 marks] 4. Using Laplace transformation solves the following: y'' y' 8y 0, y(0) 3, y'(0) 6 [0 marks] Solve the differential equation: cos y( sin cos ) [0 Marks] Solve the differential equation: ( y y e y y) ( y e y 3 ) 0 [0 Marks] Reputed Institute for IAS, IFoS Eams Page

3 7. Find the constant a so that ( y) a is the integrating factor of (4 y 6 y) ( 9y 3 ) 0 and hence solve the differential equation [ Marks] 8. (i) s 5s Obtain Laplace Inverse transform of ln e s s 5 (ii) Using Laplace transform, solve y" y t, y(0), y'(0) [6+6= Marks] 9. Solve the differential equation py p where p [3 Marks] d y 3 d y d y Solve 6 4 4y cos(log ) 4 3 e [3 Marks] 04. Justify that a differential equation of the form: y f ( y ) yf ( y ) 0 where f ( y ) is an arbitrary function of ( y ), is not an eact differential equation and y is an integrating factor for it. Hence solve this differential equation for f ( y ) ( y ) [0 Marks]. Find the curve for which the part of the tangent cut-off by the aes is bisected at the point of tangency [0 Marks] 3. Solve by the method of variation of parameters: 5y sin (0 Marks] d y d y Solve the differential equation: 3 8y 65cos 3 log e [0 Marks] 5. d y Solve the following differential equation: ( ) y e, when e is a solution to its corresponding homogeneous differential equation. M, y N, y 0, to have an 6. Find the sufficient condition for the differential equation integrating factor as a function of ( y). What will be the integrating factor in that case? Hence find the integrating factor for the differential equation of ( y) ( y y) 0 and solve it. d y 7. Solve the initial value problem y 8e t sin t, y(0) 0, y'(0) 0 by using Laplace transform. dt [0 Marks] If y is a function of, such that the differential coefficient is equal tocos y sin y. Find out a relation between and y, which is free from any derivative / differential. [0 Marks] 9. Obtain the equation of the orthogonal trajectory of the family of curves represented by n r asin n, ( r, ) being the plane polar coordinates. [0 Marks] Solve the differential equation(5 6 y ) 6y 0 Reputed Institute for IAS, IFoS Eams Page 3

4 3. Using the method of variation of parameters, solve the differential equation a y sec a 3. d y Find the general solution of the equation y ln sin(ln ) 33. By using Laplace transform method, solve the differential equation( D n ) asin( nt ), d D subject to the initial conditions 0 and 0, at t 0, in which anand, are dt dt constants. ( / y) ye 34. Solve y y ( e ) e ( / ) ( / y) 0 [ Marks] 35. Find the orthogonal trajectory of the family of curves y a [ Marks] 36. Using Laplace transforms, solve the initial value problem y y y e y y '' ' t, (0), '(0) [ Marks] 37. Show that the differential equation (y log y) ( y y ) 0 is not eact. Find an integrating factor and hence, the solution of the equation [0 Marks] 38. Find the general solution of the equation y''' y'' 6 [0 Marks] 39. Solve the ordinary differential equation ( ) y'' ( ) y' y ( 3) [0 Marks] Obtain the solution of the ordinary differential equation 4 y, if y (0) [0 Marks] 4. Determine the orthogonal trajectory of a family of curves represented by the polar equation r a( cos ), ( r, ) being the plane polar coordinates of any point. [0 Marks] 4. Obtain Clairaut s form of the differential equation y y a. Also find its general solution 43. Obtain the general solution of the second order ordinary differential equation y'' y' y e cos, where dashes denote derivatives w.r.t. 44. Using the method of variation of parameters, solve the second order differential equation 4y tan 45. Use Laplace transform method to solve the following initial value problem: d t e, (0) and dt dt dt t0 Reputed Institute for IAS, IFoS Eams Page 4

5 Consider the differential equation y', 0 where is a constant. Show that (i) If ( ) is any solution and ( ) ( ) e, then ( ) is a constant; (ii) If 0, then every solution tends to zero as [ Marks] 47. Show that the differential equation (3 y ) y( y 3) y' 0 admits an integrating factor which is a function of ( y ). Hence solve the equation [ Marks] 48. Verify that ( M Ny) d log e( y) ( M Ny) d log e( / y) M N. Hence show that- (i) If the differential equation M N 0 is homogeneous, then( M Ny ) is an integrating factor unless M Ny 0; (ii) If the differential equation M N 0 is not eact but is of the form f ( y) y f ( y) 0 then( M Ny ) is an integrating factor unless M Ny 0; [0 Marks] 49. Use the method of undermined coefficients to find the particular solutions of y'' y sin ( ) e and hence find its general solution. [0 Marks] Find the Wronskian of the set of functions: 3, 3 on the interval [, ]and determine whether the set is linearly dependent on[, ] [ Marks] 5. Find the differential equation of the family of circles in the y- plane passing through (, ) and (, ) [0 Marks] 5. Find the inverse Laplace transform of 53. Solve : y ( y) 3y y 4y 3, y(0) s F( s) n [0 Marks] s s 008 [0 Marks] Solve the differential equation y ( y ) 0 [ Marks] 55. Use the method of variation of parameters to find the general solution of y'' 4 y' 6y sin [ Marks] 56. Using Laplace transform, solve the initial value problem '' 3 ' 4 3 t y y y t e, y (0), y '(0) 57. Solve the differential equation 3 y y y '' 3 ' sin(ln ) 58. Solve the equation y p yp 0, where p Reputed Institute for IAS, IFoS Eams Page 5

6 59. Solve the ordinary differential equation 60. Find the solution of the equation y 007 y [ Marks] cos3 3 sin3 sin 6 sin 3,0 y 4 [ Marks] 6. Determine the general and singular solutions of the equation y a, a being a constant Obtain the general solution of [ D 6D D 8] y e e, 4 where D 63. d y 3 Solve the equation 3 3y 64. Use the method of variation of parameters to find the general solution of the equation d y 3 y e Find the family of curves whose tangents form an angle with the hyperbolas y c, c 0 4 [ Marks] Solve the differential equation y e y 0 [ Marks] 67. tan y Solve: ( y ) ( e ) 0 p py y y 0 using the substitution y u and y v and find its 68. Solve the equation singular solution, where p Solve the differential equation d y Solve the differential equation( D D ) y e tan, D by the method of variation of parameters Find the orthogonal trajectory of the family of co-aial circles y g c 0, where g is the parameter. [ Marks] Reputed Institute for IAS, IFoS Eams Page 6

7 7. Solve: y ( y y ) [ Marks] 73. Solve the differential equation: ( ) D ( ) D ( ) D ( ) y ( ) 74. Solve the differential equation: ( y )( p) ( y)( p)( yp) ( yp) 0 where p, by reducing it to Clairaut s form by using suitable substitution. 75. Solve the differential equation (sin cos ) y'' sin y' ysin 0 given that y sin is a solution of this equation. 76. Solve the differential equation y'' y' y log, 0 by variation of parameters Find the solution of the following differential equation y cos sin [ Marks] 78. Solve: y( y y ) ( y y ) 0 [ Marks] Solve: ( D 4D 5) y e ( cos ) 80. Reduce the equation( p y)( py ) p, where p to Clairaut s equation and hence solve it. 8. Solve: ( ) ( 5) y ( ) e 8. Solve the following differential equation: ( d y ) 4 ( ) y Show that the orthogonal trajectory of a system of confocals ellipses is self orthogonal [ Marks] 84. Solve: ylog y ye [ Marks] Solve ( D D) 4( e cos ), where D. 86. Solve the differential equation( p y )( p y) ( P ), where p, by reducing it to Clairaut s form using suitable substitutions 87. Solve ( ) y'' ( ) y' y sinlog( ) Solve the differential equation y'' 4 y' 6y sec by variation of parameters. Reputed Institute for IAS, IFoS Eams Page 7

8 Solve : 3y y [ Marks] 90. d y Find the values of for which all solutions of 3 y 0tend to zero as. [ Marks] 9. Find the value of constant such that the following differential equation becomes eact. y y (e 3 y ) (3 e ) 0. Further, for this value of, solve the equation. 9. y 4 Solve : y 6 d y 93. Using the method of variation of parameters, find the solutions of y e sin with y(0) 0 and Solve : ( D )( D D ) y e where D 00 t e, 0 t 95. A continuous function yt () satisfies the differential equation If t 3 t, t 5 y(0) e find y () [ Marks] 96. d y Solve : 3y log e [ Marks] 97. y y(log e y) Solve : loge y 98. Find the general solution of ayp ( b) p y 0, a Solve: ( D ) y 4 cos given that y Dy D y 0 and 3 Dy when Using the method of variation of parameters, solve 4y 4tan 000 d y 0. Show that 3 4 8y 0 has an integral which is a polynomial in. Deduce the general solution. [ Marks] d y 0. Reduce P Qy R, where P, Q, R are functions of, to the normal form. Hence solve d y 4 (4 ) y 3e sin Reputed Institute for IAS, IFoS Eams Page 8

9 03. Solve the differential equation y ap ap. Find the singular solution and interpret it geometrically 04. Show that(4 3y ) (3 y ) 0 represents a family of hyperbolas with a common ais and tangent at the verte 05. d y Solve y ( ) by the method of parameters 06. Solve the differential equation 3 d y d y 999 y y y y Reputed Institute for IAS, IFoS Eams Page 9 / [0 Marks] 07. Solve 3 4 y e cos 3 [0 Marks] 08. By the method of variation of parameters solve the differential equation a y sec( a) 998 [0 Marks] Solve the differential equation: y y e [0 Marks] 0. Show that the equation: (4 3y ) (3 y ) 0 represents a family of hyperbolas having as asymptotes the lines y 0, y 0. [0 Marks]. Solve the differential equation: y 3p 4p [0 Marks]. d y 4 Solve the differential equation: 5 6 y e ( 9) [0 Marks] 3. d y Solve the differential equation: y sin [0 Marks] d y d y Solve the differential equation: y 0 3 [0 Marks] Solve the initial value problem, y (0) 0 3 y y [0 Marks] 6. Solve( y 3 y) ( y 3 y) 0 [0 Marks] 7. Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its radius originally is 3mm, and one hour later has been reduced to mm. find an epression for the radius of the rain drop at any time. [0 Marks] d y d y d y Solve 6 6 0e sin 4 3 [0 Marks]

10 9. Make use of the transformation y( ) u( )sec to obtain the solution of y'' y'tan 5y 0, y(0) 0, y'(0) 6 [0 Marks] 0. Solve ( ) d y 6( ) 6y 8( ), y(0) 0, y'(0) [0 Marks] 3. Solve : 996. Find the curves for which the sum of the reciprocals of the radius vector and polar sub tangent is constant. [0 Marks]. Solve : ( y p) yp, p [0 Marks] ysin y cos 0 [0 Marks] 4. d y 0y 37sin3 0, if it is given that y 3 and 0 when 0 [0 Marks] d y d y d y Solve : 3 3e 4sin 4 3 [0 Marks] d y d y Solve : 3 y log 3 [0 Marks] Determine a family of curves for which the ratio of the y- intercept of the tangent to the radius vector is a constant. [0 Marks] 8. Solve( 3y 7) ( 3 y 8) y 0 [0 Marks] 9. Test whether the equation ( y) ( y y ) 0 is eact and hence solve it. [0 Marks] d y d y Solve y 0 3 [0 Marks] 3. Determine all real valued solutions of the equations y''' iy'' y' iy 0, y' [0 Marks] 3. Find the solution of the equation 4y 8cos, given that 0 [0 Marks] 3 sin y cos y Solve: [0 Marks] 34. Show that if P Q Q y is a function of only, say, ( ), f then ( ) F( ) e f is an integration factor of P Q 0 [0 Marks] Reputed Institute for IAS, IFoS Eams Page 0

11 35. Find the family of curves whose tangent from an angel with the hyperbola y c [0 Marks] 4 d y Transform the differential equation cos sin ycos cos into one having z as independent variable where z sin and solve it. [0 Marks] d g 37. If ( ) 0 a ( aband, g being positive constants) and a' and 0 when 0, dt b dt t g show that a ( a' a)cost t [0 Marks] b 38. Solve ( D 4D 4) y 8 e sin where D [0 Marks] Determine the curvature for which the radius of curvature is proportional to the slope of the tangent. [0 Marks] 40. y Show that the system of co focal conics is self orthogonal. a b [0 Marks] 4. Solve y cos y log sin y 0 [0 Marks] 4. Solve 43. Solve d y y y 0 cos [0 Marks] dt y a t and discuss the nature of solution as [0 Marks] 4 / Solve ( D D ) y e cos 99 0 [0 Marks] 45. By eliminating the constants abobtain, the differential equation for which y ae be is a solution [0 Marks] 46. Find the orthogonal trajectory of the family of semi cubical parabolas ay 3, where a is a variable parameter. [0 Marks] 47. Show that (4 3y ) (3 y ) 0 represents hyperbolas having the following lines as asymptotes y 0, y 0 [0 Marks] 48. Solve the following differential equation y( y) ( y) 0 [0 Marks] 49. Find the curves for which the portion of y- ais cut off between the origin and the tangent varies as the cube of the abscissa of the point of contact. [0 Marks] 50. Solve the following differential equation: ( D 4) y sin, given that when 0, then y 0 and 5. 3 Solve : ( D ) y e cos [0 Marks] [0 Marks] 5. Solve : ( D D 4) y [0 Marks] Reputed Institute for IAS, IFoS Eams Page