Previous Years Questions ( ) Segment-wise

Transcription

1 Institute for IAS/ IFoS/ CSIR/ GATE Eaminations Preious Years Questions (98 ) Segment-wise Orinary Differential Equations an Laplace Transforms (Accoring to the New Syllabus Pattern) Paper - I 98 Sole + ( ) y= Sole (y +yz) +(z+z ) + (y y) z = Sole the equation + + y = t by the metho t t of Laplace transform, gien that y = when t =, y = when t = Sole + y = sec 98 u Using the transformation y = k, sole the equation y + (+k) y +y = Sole the equation ( D + ) = tcost, gien that = = by the metho of Laplace transform 985 Consier the equation y +5y = Fin that solution φ of the equation which satisfies φ () = φ () Use Laplace transform to sole the ifferential t equation + = e, ' = t () =, () = such that For two functions f, g both absolutely integrable on (, ), efine the conolution f * g If L(f), L(g) are the Laplace transforms of f, g show that L (f * g) = L(f) L(g) Fin the Laplace transform of the function nπ t < (n+ ) π f() t = (n+) π t (n+) π n =,,, Sole the equation ( ) = y + e If f(t) = t p, g(t) = t q for t > but f(t) = g(t) = for t, an h(t) = f * g, the conolution of f, g show that an p, q are Γ( p) Γ( q) p+ q ht () = t ; t Γ ( p+ q) positie constants Hence euce the formula Γ( p) Γ( q) B( pq, ) = Γ ( p + q) 988 Sole the ifferential equation = e sin Show that the equation (+7y+) + (7+y+) = represents a family of cures haing as asymptotes the lines +y =, +y+= Obtain the ifferential equation of all circles in a plane n the form + = 989 Fin the alue of y which satisfies the equation (y y e ) + y =; gien that y= when = Proe that the ifferential equation of all parabolas lying in a plane is = Sole the ifferential equation = + 6 Hea Office: 5-6, Top Floor, Mukherjee Tower, Dr Mukherjee Nagar, Delhi-9 Branch Office: 5/8, Ol Rajener Nagar Market, Delhi-6 Ph: , 99999, imsims@gmailcom,

2 99 (a) If the equation λ n +a λ n ++ a n = (in unknown λ) has istinct roots λ,λ, λ Show that the constant coefficients of ifferential equation n n + a n n ++ a n an b + = has the most general solution of the form y = c () + c e λ + c e λ + + c n e λ n where c, c c n are parameters what is c ()? (b) Analyse the situation where the λ equation in (a) has repeate roots Sole the ifferential equation + + y = is eplicit form If your answer contains imaginary quantities, recast it in a form free of those Show that if the function t f() t can be integrate y (wrt t ), then one can sole f ( ) =, for any gien f Hence or otherwise y+ + = y+ 6 Verify that y = (sin ) is a solution of ( ) = Fin also the most general solution 99 Institute for IAS/ IFoS/ CSIR/ GATE Eaminations n If the equation M + N = is of the form f (y) y + f (y) =, then show that is an M Ny integrating factor proie M Ny Sole the ifferential equation ( +y ) + y = Gien that the ifferential equation ( y +y) ( y ) = has an integrating factor of the form k y k, fin its general solution Sole my sin = m Sole the ifferential equation y = e Sole the ifferential equation 5 y e when = =, gien that y = an =, 99 By eliminating the constants a, b obtain the ifferential equation of which y = ae +be + is a solution Fin the orthogonal trajectories of the family of semicubical parabolas ay =, where a is a ariable parameter Show that (+y+) + (+y+) = represents hyperbolas haing the following lines as asymptotes +y =, +y+ = (998) Sole the following ifferential equation y (+y) + ( y) = Sole the following ifferential equation (D +) y = sin gien that when = then y = an = Sole (D )y = e + cos Sole ( D +D ) y = 99 Show that the system of confocal conics y a + λ b + λ + = Sole = Sole y + + cos y is self orthogonal + { + log sin y} wy + = a coswt an iscuss the nature of solution as w w Sole (D +D +) y = e cos 99 Sole + sin y = cos y Show that if P Q is a function of only Q y say, f(), then F( ) f ( ) = is an integrating e factor of P + Q = Hea Office: 5-6, Top Floor, Mukherjee Tower, Dr Mukherjee Nagar, Delhi-9 Branch Office: 5/8, Ol Rajener Nagar Market, Delhi-6 Ph: , 99999, imsims@gmailcom,

3 Fin the family of cures whose tangent form angle Π with the hyperbola y = c Transform the ifferential equation 5 cos+ sin ycos = cos into one haing z an inepenent ariable where z = sin an sole it If t Institute for IAS/ IFoS/ CSIR/ GATE Eaminations g + ( a ) =, (a, b an g being positie b constants) an = a an t = when t=, show that g = a+ ( a a)cos t b Sole (D D+) y = 8 e sin, where, 995 Sole ( +y 7) ( +y 8) y = D = Test whether the equation (+y) (y y ) = is eact an hence sole it Sole + + y = + (998) Determine all real alue solutions of the equation y iy + y iy =, y = Fin the solution of the equation y + y = 8cos gien that y = an = when = 996 Sole (y p) = yp ; p = Sole y sin (+y + cos ) = Sole + +y+7 sin = Fin the alue of y when = Π, if it is gien that y= an = when = Sole Sole + = + e + sin y = + log 997 Sole the initial alue problem = y+ y, y()= Sole ( y + y) + ( y + y) = Sole = e sin Make use of the transformation y() = u() sec to obtain the solution of y y tan+ 5y = ; y()=; y () = 6 Sole (+) 6 (+) 6y + = 8 (+) ; y() = an () = 998 Sole the ifferential equation y = ye Show that the equation (+y+) + (+y+) = represents a family of hyperbolas haing as asymptotes the lines +y = ; +y+= (99) Sole the ifferential equation y = p + p Sole 5 6 y e ( 9) + = + Sole the ifferential equation + + y = sin 999 Sole the ifferential equation + y y = y + y Sole + y = e + cos By the metho of ariation of parameters sole the ifferential equation Show that ay sec( a) + = + 8y = has an integral Hea Office: 5-6, Top Floor, Mukherjee Tower, Dr Mukherjee Nagar, Delhi-9 Branch Office: 5/8, Ol Rajener Nagar Market, Delhi-6 Ph: , 99999, imsims@gmailcom,

4 which is a polynomial in Deuce the general solution Reuce + P + Qy = R, where P, Q, R are functions of, to the normal form Hence sole ( ) y e sin + = Sole the ifferential equation y = a p+ap Fn the singular solution an interpret it geometrically Show that (+y+)+(+y+) = represents a family of hyperbolas with a common ais an tangent at the erte Sole y= ( ) + by the metho of ariation of parameters A continuous function y(t) satisfies the ifferential equation t + e, t < = t + t t t, 5 If y() = e, fin() Sole Sole Institute for IAS/ IFoS/ CSIR/ GATE Eaminations y = log e y y(log e y) + loge y = Fin the general solution of ayp +( b) p y=, a>o Sole (D +) y = cos gien that y=dy=d y= an D y = when = Using the metho of ariation of parameters, sole y tan + = Sole y y + = Fin the alues of λ for which all solutions of + λy = ten to zero as Hea Office: 5-6, Top Floor, Mukherjee Tower, Dr Mukherjee Nagar, Delhi-9 Branch Office: 5/8, Ol Rajener Nagar Market, Delhi-6 Ph: , 99999, imsims@gmailcom, Fin the alue of constant λ such that the following ifferential equation becomes eact y y ( e + y ) + ( + λe ) = Further, for this alue of λ, sole the equation Sole + y+ = y 6 Using the metho of ariation of parameters, fin the solution of y() = an = + y = e sin with = Sole (D ) (D D+) y = e where D = Show that the orthogonal trajectory of a system of confocal ellipses is self orthogonal Sole ylog y ye + = Sole (D 5 D) y = (e +cos + ), where D = Sole the ifferential equation (p + y ) (p + y) = (p + ) where p =, by reucing it to Clairaut s form using suitable substitutions Sole ( + ) y + ( + ) y + y = ( + ) sin log Sole the ifferential equation + = by ariation of y y 6y sec parameters Fin the solution of the following ifferential equation ycos sin + = Sole y(y+ y ) + (y y ) = Sole ( ) D D 5 y = e ( + cos )

5 Reuce the equation (p y) (py+) = p where p = to Clairaut s equation an hence sole it Sole (+) Institute for IAS/ IFoS/ CSIR/ GATE Eaminations (+ 5) + y = ( + ) e Sole the following ifferential equation ( ) ( ) y + = 5 Fin the orthogonal trejectory of a system of co-aial circles +y +g+c=, where g is the parameter Sole y y y = Sole the ifferential equation (+) D + (+) D (+) D +(+)y = + Sole the ifferential equation ( +y ) (+p) (+y) (+p) (+yp) + (+yp) =, where p =, by reucing it to Clairaut s form by using suitable substitution Sole the ifferential equation (sin cos ) y siny + ysin= gien that y = sin is a solution of this equation Sole the ifferential equation + = log, > by ariation of y y y parameters 6 Fin the family of cures whose tangents form an angle Π with the hyperbolas y=c, c > Sole the ifferential eqaution ( ) y + e y = + + = Sole ( ) ( ) tan y y e Sole the equation p + yp ( + y) + y = using the substituion y = u an y= an fin its singular solution, where p = Sole the ifferential equation + + y = + Hea Office: 5-6, Top Floor, Mukherjee Tower, Dr Mukherjee Nagar, Delhi-9 Branch Office: 5/8, Ol Rajener Nagar Market, Delhi-6 Ph: , 99999, imsims@gmailcom, Sole the ifferential equation ( D D+ ) y = e tan, where D =, by the metho of ariation of parameters 7 Sole the orinary ifferential equation cos π ysin= sin6+ sin, < < Fin the solution of the equation y y + = Determine the general an singular solutions of the equation y = + a + a being a c o n s t a n t Obtain the general solution of D 6D + D 8 9 y = e + e, where D = Sole the equation + y = Use the metho of ariation of parameters to fin the general solution of the equation + + y = e 8 Sole the ifferential equation ( ) y+ + y = Use the metho of ariation of parameters to fin the general solution of y y + 6 y = sin Using Laplace transform, sole the initial alue t problem y y + y = t + e

6 with y() =, y () = Sole the ifferential equation + = sin(ln ) + y y y Sole the equation y p+ yp = where 9 Fin the Wronskian of the set of functions {, } Institute for IAS/ IFoS/ CSIR/ GATE Eaminations p = on the interal [, ] an etermine whether the set is linearly epenent on [, ] Fin the ifferential equation of the family of circles in the y-plane passing through (, ) an (, ) Fin the inerse Laplace transform of s + Fs () = ln s+ 5 Sole: y ( y) =, y() = y y y Consier the ifferential equation y = α, > where α is a constant Show that (i) if φ() is any solution an Ψ() = φ() e α, then Ψ() is a constant; (ii) if α <, then eery solution tens to zero as Show that the iffrential equation ( ) + ( ) = y y y y amits an integrating factor which is a function of (+y ) Hence sole the equation Verify that ( M+ Ny) ( log e( y) ) + ( M Ny) (log e( y)) = M + N Hence show that (i) if the ifferential equation M + N = is homogeneous, then (M + Ny) is an integrating factor unless M+ Ny ; (ii) if the ifferential equation M+ N = is not eact but is of the form f ( y) y+ f ( y) = then (M Ny) is an integrating factor unless M Ny Show that the set of solutions of the homogeneous linear ifferential equation y + py ( ) = on an interal I = [ ab, ] forms a ector subspace W of the real ector space of continous functions on I what is the imension of W? Use the metho of unetermine coefficiens to fin the particular solution of y + y = + + e sin ( ) an hence fin its general solution Obtain the soluton of the orinary ifferential equation ( y ), = + + if y() = Determine the orthogonal trajectory of a family of cures represente by the polar equation r = a( cosθ), (r, θ) being the plane polar coorinates of any point Obtain Clairaut s orm of the ifferential equation y y + y = a Also fin its general solution Obtain the general solution of the secon orer orinary ifferential equation y y + y= + e cos, where ashes enote eriaties wr to Using the metho of ariation of parameters, sole the secon orer iffereifferential equation y tan + = Use Laplace transform metho to sole the following initial alue problem: t e t t t t= + =, () = an = Hea Office: 5-6, Top Floor, Mukherjee Tower, Dr Mukherjee Nagar, Delhi-9 Branch Office: 5/8, Ol Rajener Nagar Market, Delhi-6 Ph: , 99999, imsims@gmailcom,