Tiêu đề MP DPP Sheet 04.pdf ✅

1 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 IIT JAM PHYSICS 2025 (Online Batch) SECTION: MATHEMATICAL PHYSICS Daily Practice Problem (DPP) Sheet 4: DIFFERENTIAL EQUATIONS (Linear Non-Homogeneous ODE with constant coefficients, Cauchy Euler ODE) Q.1. Solve the differential equations: (i) 2 2 6 cosh2 d y dy x y e x dx dx    [Ans. 3 2 3 1 2 1 1 10 8 x x x x y c e c e xe e       ] (ii) 2 2 2 2 sinh d y dy y x dx dx    [Ans.   1 2 1 1 cos sin 10 2 x x x y e c x c x e e       ] (iii) 2 2 d y g g y L dx l l   given that     ' y a y 0 , 0 0   [Ans.   cos g y a L x L l    ] (iv)  3 3 1 16 x D y e   [Ans.   2 3 1 2 3 2 x x y c c x c x e e     ] (v) 2 2 2 2 3 4 2 d y dy y x x dx dx     [Ans.   3 4 2 1 2 23 23 1 cos sin 8 28 13 4 4 32 x y e c x c x x x             ] (vi) 2 2 2 2 4 d y dy x x dx dx     [Ans. 3 1 2 4 3 x x y c c e x      ] (vii) 2 2 4 2cos cos3 d y y x x dx   [Ans. 1 2 1 cos2 sin 2 cos4 sin 4 12 4 x y c x c x x x     ] (viii) 2 3 2 2 4 4 d y dy x y x e dx dx    [Ans.   5 2 2 1 2 20 x x x y c c x e e    ] (ix) 2 2 2 sin d y dy x y x e x dx dx    [Ans.     1 2 sin 2cos x x y c c x e e x x x     ] (x) 2 2 4 2 2 4 d y dy x x y x dx dx    [Ans. 4 4   2 1 ln 5 c x y c x x x    ] (xi)   2 2 2 2 sin ln d y dy x x y x dx dx    [Ans.       2 1 2 1 cos ln sin ln sin ln 3 y c x c x x    ] (xii) 2 2 2 ln d y dy x x y x dx dx    [Ans. y C C x x x      1 2 ln ln 2  ]
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 PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) Q.2. The particular integral of the differential equation 2 2 4 2 d y x y dx   can be written as (a)  2 1 2 ln 2 4 x  (b)  2 1 2 ln 2 4 x  (c)  2 1 2 ln 2 2 x  (d)  2 1 2 ln 2 16 x  Q.3. The particular integral of the differential equation 3 2 3 2 6 11 6 d y d y dy x y e dx dx dx     can be written as (a) 1 2 x e (b) 1 2 x xe (c) x xe (d) x e Q.4. The particular integral of the differential equation     4 2 2 4 D a D a y ax    2 8cos , where a is a real con- stant, is of the form     2 A Bx ax  cos . Which of the following is CORRECT? (a) A B   0, 0 (b) A B   0, 0 (c) A B   0, 0 (d) A B   0, 0 Q.5. The particular integral of the differential equation   2 D D y x   2 , is of the form   3 2 Ax Bx Cx   . Which of the following is CORRECT? (a) 1 , 1, 2 3 A B C    (b) 1 , 1, 2 3 A B C     (c) 1 , 1, 2 3 A B C     (d) 1 , 1, 2 3 A B C     Q.6. The particular integral of the differential equation y y y x " ' 3 5cos 2 3       is: (a) 2cos 2 3 sin 2 3  x x       (b) 2sin 2 3 cos 2 3  x x       (c) sin 2 3 2cos 2 3  x x       (d) 2sin 2 3 cos 2 3  x x       Q.7. Consider the following 2nd order differential equation: 2 2 2 4 3 2 3 d y dy y x x dx dx     The particular solution of the differential equation is (a) 2    2 2x x (b) 2   2x x (c) 2 2x x  (d) 2    2 2x x Q.8. The particular integral of 4 4 cos cosh is: d y y x x dx   (a) 1 sinh cos 5 y x x   (b) 1 cosh cos 5 y x x   (c) 1 sinh sin 5 y x x   (d) 1 cosh sin 5 y x x   Q.9. The solution of the differential equation: 4 2 2 4 2 3 108 d y d y x 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 (a)     4 2 1 2 3 4 C C x C x C x x x K       sin 3 cos 3 3 12 (b)     4 2 1 2 3 4 C C x C x C x x x K       sin 3 cos 3 5 8 (c)     4 2 2 3 4 C x C x C x x x K      sin 3 cos 3 5 8 (d)     2 4 2 2 3 4 C x C x C x x x K      sin 3 cos 3 3 12 Q.10. If   4x e V x is the particular integral of the following differential equation   2 10 20 4 2 8 16 2 11 21 d y dy x y x x x e dx dx      Then the V x  will be of the form (a) 10 20 Ax Bx Cx   (b)   2 11 21 Ax Bx Cx   (c)   3 12 22 Ax Bx Cx   (d) none of these Q.11. Consider the differential equation 2 2 2 4 d y dy x x y dx dx   with boundary conditions y 0 0   and y 1 1   . The complete solution of the differential equation will be (a) 2 x (b) 3 x (c) x (d) 2 x  Q.12. Consider the ordinary differential equation: 2 2 2 2 2 0 d y dy x x y dx dx    subjected to the conditions y y 1 0 and 2 2      . The value of y 3 is (a) 3 (b) 6 (c) 9 (d) 12 Q.13. The general solution of differential equation 2 4 " 8 ' 9 0 is: x y xy y    (a) 5 /2 3 /2 1 2 x x C e C e  (b) 3 /2 3 /2 1 2 x x C e C e  (c)   3 2 1 2 C C x x  log (d) 3 3 2 2 C x C x 1 2   Q.14. The complementary function of the differential equation   3 3 2 2 x D x D y x    2 2 10ln is (a) C x x C x C x 1 2 3     cos ln sin ln       (b) C x x C x C x 1 2 3 / cos ln sin ln           (c) C x x C x C x 1 2 3 / cos sin     (d) C x x C x C x 1 2 3    cos sin  Q.15. The particular integral of the differential equation 2 2 2 4 3 5 10 d y dy x x y dx dx x     is (a) 1/ 5 x (b) 1/ 5 1/  x (c) 2 1/  x (d) 2 1/  x Q.16. The complementatry function of the differential equation         2 2 2 1 1 2 3 2 4 d y dy x x x x dx dx       is (a) A B x    1 (b)  x 1 A Be   (c)  x 1 A Be   (d) A B x   ln 1  
4 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 PART - B: Numerical Answer Type (NAT) Questions Q.17. Suppose y x x x p    cos 2  is a particular solution of y ay x " 4sin 2    , then the constant ‘a’ is equal to _____________________________________________ [Your answer should be AN INTEGER] Q.18. If C C x x 1 2  ln  is the general solution of the differential equation   2 2 2 0 0 d y dy x kx y x dx dx     then k is equal to _______________________________ [Your answer should be AN INTEGER] PART - C: Multiple Select Questions (MSQ) Q.19. Consider a forced harmonic oscillator which obeys the differential equation : 2 2 sin d y y t dt   Which of the following is the solution of the differential equation with the initial condition y (0) = 0? (a) y t t    6sin (b)   12sin cos 2 t y t t t   (c)   12sin cos 2 t y t t t   (d)   6sin cos 2 t y t t t   Q.20. Which of the following statement is/are CORRECT about the differential equation   2 D D y x x    2 1 cos ? (a) C.F. is   1 2 x c c x e   (b) C.F. for this equation is 2 1 2 x x c e c e   (c) P.I. is   1 1 sin cos 2   x x x     (d) P.I. is   1 1 sin cos 2   x x x     Answer Key PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) 2. (b) 3. (b) 4. (c) 5. (d) 6. (d) 7. (a) 8. (b) 9. (a) 10. (c) 11. (a) 12. (b) 13.(c) 14. (b) 15. (c) 16. (d) PART - B: Numerical Answer Type (NAT) Questions 17. (4) 18. (-1) PART - C: Multiple Select Questions (MSQ) 19. (c,d) 20. (a,c)