Title IIT-JAM Differential Equation DPP Sheet 03.pdf ✅

2 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.4. The solution of the differential equation '' y y   4 0 subjected to the condition y 0 1,     ' y 0 2  is (a) sin 2 cos 2 x x  (b) sin 2 cos 2 x x  (c) 1 sin 2 cos 2 2 x x  (d) 1 sin 2 cos 2 2 x x  Q.5. The solution of the differential equation 2 2 2 0 d x dx x dt dt    , which satisfies x 0 3   and does not blow up as t  , will be (a) 3 t x e   (b) 2 4 t t e e x    (c) 2 3 t x e  (d) 2 2 t t x e e     Q.6. The general solution of the differential equation 5 4 3 2 5 4 3 2 3 3     0 d y d y d y d y dx dx dx dx is (a)   2 1 2 3 4 5 x c c x c c x c x e     (b) 2 2 3 1 2 3 4 5     x x x c x c x c e c e c e (c)  1 2 3 4 5      x c c c c c e (d) None of these Q.7. The general solution of the differential equation 3 2 3 2 9 27 27 0 d y d y dy y dx dx dx     is (a)     2 3x y x A Bx Cx e    (b)     2 3x y x A Bx Cx e    (c)   3 3 3 x x x y x Ae Be Ce    (d)   3 3 3 x x x y x Ae Be Ce       Q.8. The general solution of the differential equation 4 4 4 d y m y dx  is (a)   1 2 3 4 y x C mx C mx C mx C mx     cos sin cosh sinh (b)   1 2 3 4 cos sin   mx y x C mx C mx C C x e     (c)     2 3 1 2 3 4 mx y x C C x C x C x e     (d)    1 2 3 4  cosh sinh mx y x C C x e C mx C mx     Q.9. The solution of the ordinary differential equation 2 2 0 d y dy p qy dx dx    is 3 1 2 x x y C e C e     . The value of p and q are respectively (a) 3, 3 (b) 3, 4 (c) 4, 3 (d) 4, 4 Q.10. If 6 and -9 are the roots of the auxiliary equation, then the differential equation can be represented as (a) 2 2 4 12 0 d y dy y dx dx    (b) 2 2 4 12 0 d y dy y dx dx    (c) 2 2 3 54 0 d y dy y dx dx    (d) 2 2 3 54 0 d y dy y dx dx   
3 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.11. A function n(x) satisfies the differential equation     2 2 2 0 d n x n x dx L   , where L is a constant and subjected to the conditions n K 0  and n  0 . The solution of the equation will be (a)   x L/ n x Ke  (b)   x L/ n x Ke  (c)   x L / n x Ke  (d)   x L / n x Ke  Q.12. The solution to the differential equation 2 2 d u du k dx dx  , where k is constant, subjected to the boundary conditions     0 u u L u 0 0,   , is (a) 0 x u u L  (b) 0 1 1 kx kL e u u e          (c) 0 1 1 kx kL e u u e            (d) 0 1 1 kx kL e u u e          Q.13. Consider the differential equation: 2 2 0 d y y dt   with initial conditions y 0 1,   0 1 t dy dt   for 0 t  . The solution attains a maximum for t equal to (a) 4 t   (b) 2 t   (c) t  0 (d) t   Q.14. The differential equation 2 2 0 d y y dx   satisfying the conditions y y 0 1, 0      , has (a) a unique solution (b) infinite no.of solutions (c) no solution (d) none of these Q.15. A solution of the ordinary differential equation 2 2 5 6 0 d y dy y dx dx    is such that     3 3 1 0 2 and 1 e y y e    . The value of   0 dy dx is (a) 1 (b) e (c) e2 (d) -3 Q.16. If the characteristic equation of the differential equation 2 2 2 0 d y dy y dx dx     has two equal roots, then the values of  are (a) +1, -1 (b) 0, -1 (c) 0, 1 (d) +1/2, -1/2 Q.17. Consider the following differential equation:       2 2 8 6 0 with 0 1and Lim 0 x d y i y y y x dx       The value of y 2  is (a) 6 e  (b) 6 e   (c) 6 e   (d) 6 e   