Differential Equations Class 12 Notes

• Example:
• $\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}=\sin x$
• has order 2.
• To read the degree, first clear radicals in the highest derivative.
• A general solution contains as many arbitrary constants as the order; a particular solution fixes them with initial conditions.

• Differentiate the general solution as many times as it has constants to eliminate them.
• Example:
• $y=A\cos x+B\sin x$
• →
• $y^{\prime \prime }=-y$
• (eliminating A and B).
• The result is a differential equation of order equal to the number of constants eliminated.

• Write
• $\frac{dy}{dx}=f(x)g(y)$
• as
• $\frac{dy}{g(y)}=f(x)dx$
• .
• Integrate both sides and add a single constant C.
• Apply the initial condition last to find the particular solution.

• To test, put y = vx: every term in x and y becomes a power of x times a function of v.
• After solving, substitute v = y/x back to express the answer in x and y.
• Homogeneous equations frequently appear in JEE as short-answer questions.

• Solution:
• $y\cdot e^{\int Pdx}=\int Q{e}^{\int Pdx}dx+C$
• .
• Bring every equation into the standard 'dy/dx + Py = Q' form before applying the formula.
• Common exams test e^x, sin x or xᵏ as Q with an easily integrated P.

Example: Solve $\frac{dy}{dx}=xy$ with $y(0)=1$.

Solution: Separating variables: $\frac{dy}{y}=xdx$. Integrating: $\ln y=\frac{x^2}{2}+C$, so $y=A{e}^{x^2/2}$. Using $y(0)=1$ gives $A=1$, hence $y=e^{x^2/2}$.

Exercises (NCERT Q1–Q80)

• 1) Determine the order and degree (if defined) of the differential equation
• 2) Determine the order and degree of
• 3) Determine the order and degree of
• 4) Determine the order and degree of
• 5) Determine the order and degree of
• 6) Determine the order and degree of
• 7) Determine the order and degree of
• 8) Determine the order and degree of
• 9) Determine the order and degree of
• 10) Determine the order and degree of
• 11) Determine the order and degree of
• 12) Determine the order and degree of
• 13) Verify that is a solution of the differential equation
• 14) Verify that y = x² + 2x + c is a solution of y' − 2x − 2 = 0.
• 15) Verify that y = cos x + c is a solution of y' + sin x = 0.
• 16) Verify that y = √(1 + x²) is a solution of y' = xy/(1 + x²).
• 17) Verify that y = Ax (A a constant) is a solution of xy' = y, x ≠ 0.
• 18) Verify that y = x sin x is a solution of xy' = y + x√(x² − y²), x ≠ 0, 0 < y < x (with cos x > 0).
• 19) Verify that y = √(a² − x²), x ∈ (−a, a), is a solution of x + yy' = 0, y ≠ 0.
• 20) Verify that y = cos x − sin x is a solution of y'' + y = 0.
• 21) Verify that x + y = tan⁻¹ y is a solution of y²y' + y² + 1 = 0.
• 22) Verify that y = a cos x + b sin x (a, b ∈ ℝ) is the general solution of y'' + y = 0.
• 23) The number of arbitrary constants in the general solution of a differential equation of fourth order is: (A) 0 (B) 2 (C) 3 (D) 4.
• 24) The number of arbitrary constants in the particular solution of a differential equation of third order is: (A) 3 (B) 2 (C) 1 (D) 0.
• 25) Form the differential equation of the family of curves
• 26) Form the differential equation of the family
• 27) Form the differential equation of
• 28) Form the differential equation of
• 29) Form the differential equation of
• 30) Form the differential equation of the circles with centre (a, b) and radius r:
• 31) Form the differential equation of
• 32) Form the differential equation of
• 33) Form the differential equation of
• 34) Form the differential equation of
• 35) Form the differential equation of
• 36) Choose the correct answer: y = x is a particular solution of (A) y'' = xy (B) y' = xy (C) y'' − x²y' + xy = 0 (D) y'' = x²y.
• 37) Choose the correct answer: y = c₁eˣ + c₂e⁻ˣ is the general solution of (A) y'' + y = 0 (B) y'' − y = 0 (C) y'' − y' = 0 (D) y'' + y' = 0.
• 38) Solve
• 39) Solve
• 40) Solve with y(0) = 5.
• 41) Solve
• 42) Solve
• 43) Solve
• 44) Solve
• 45) Solve
• 46) Solve
• 47) Solve
• 48) Solve
• 49) Solve
• 50) Solve
• 51) Solve with y(0) = 1.
• 52) Show that is homogeneous and solve it.
• 53) Show that is homogeneous and solve it.
• 54) Show that is homogeneous and solve it.
• 55) Show that is homogeneous and solve it.
• 56) Show that is homogeneous and solve it.
• 57) Show that is homogeneous and solve it.
• 58) Show that is homogeneous and solve it.
• 59) Show that is homogeneous and solve it.
• 60) Show that is homogeneous and solve it.
• 61) Show that is homogeneous and solve it.
• 62) Show that is homogeneous; given y(1) = 1, find the particular solution.
• 63) Show that is homogeneous; given y(1) = 0, find the particular solution.
• 64) Solve
• 65) Solve
• 66) Solve
• 67) Solve
• 68) Solve
• 69) Solve
• 70) Solve
• 71) Solve
• 72) Solve
• 73) Solve
• 74) Solve
• 75) Solve (y > 0)
• 76) Solve given y(0) = 0.
• 77) Solve given y(0) = 0.
• 78) Solve given y(π/2) = 2.
• 79) Solve given y(π/2) = 0.
• 80) Choose the correct answer: the integrating factor of (−1 < y < 1) is (A) 1/(y²−1) (B) 1/√(y²−1) (C) 1/(1−y²) (D) 1/√(1−y²).

Answers at a glance

1. 1
   Q1 → Order 4; degree not defined.
2. 2
   Q2 → Order 1; degree 1.
3. 3
   Q3 → Order 2; degree 1.
4. 4
   Q4 → Order 2; degree not defined.
5. 5
   Q5 → Order 2; degree 1.
6. 6
   Q6 → Order 3; degree 2.
7. 7
   Q7 → Order 3; degree 1.
8. 8
   Q8 → Order 1; degree 1.
9. 9
   Q9 → Order 2; degree 1.
10. 10
    Q10 → Order 2; degree 1.
11. 11
    Q11 → Order 2; degree not defined.
12. 12
    Q12 → Order 2; degree 4.
13. 13
    Q13 → Yes — the function satisfies y'' − y' = 0.
14. 14
    Q14 → Yes — it is a solution (a family of parabolas).
15. 15
    Q15 → Yes — it is a solution.
16. 16
    Q16 → Yes — it is a solution.
17. 17
    Q17 → Yes — it is a solution.
18. 18
    Q18 → Yes — it is a solution.
19. 19
    Q19 → Yes — it is a solution on the given interval.
20. 20
    Q20 → Yes — it is a solution.
21. 21
    Q21 → Yes — it is a solution.
22. 22
    Q22 → Yes — y = a cos x + b sin x is the general solution.
23. 23
    Q23 → Option (D) — 4.
24. 24
    Q24 → Option (D) — 0.
25. 25
    Q25 → answer not in source — verify against the textbook
26. 26
    Q26 → answer not in source — verify against the textbook
27. 27
    Q27 → answer not in source — verify against the textbook
28. 28
    Q28 → answer not in source — verify against the textbook
29. 29
    Q29 → answer not in source — verify against the textbook
30. 30
    Q30 → answer not in source — verify against the textbook
31. 31
    Q31 → answer not in source — verify against the textbook
32. 32
    Q32 → answer not in source — verify against the textbook
33. 33
    Q33 → answer not in source — verify against the textbook
34. 34
    Q34 → answer not in source — verify against the textbook
35. 35
    Q35 → answer not in source — verify against the textbook
36. 36
    Q36 → Option (C) — y'' − x²y' + xy = 0.
37. 37
    Q37 → Option (B) — y'' − y = 0.
38. 38
    Q38 → answer not in source — verify against the textbook
39. 39
    Q39 → answer not in source — verify against the textbook
40. 40
    Q40 → answer not in source — verify against the textbook
41. 41
    Q41 → answer not in source — verify against the textbook
42. 42
    Q42 → answer not in source — verify against the textbook
43. 43
    Q43 → answer not in source — verify against the textbook
44. 44
    Q44 → answer not in source — verify against the textbook
45. 45
    Q45 → answer not in source — verify against the textbook
46. 46
    Q46 → answer not in source — verify against the textbook
47. 47
    Q47 → answer not in source — verify against the textbook
48. 48
    Q48 → answer not in source — verify against the textbook
49. 49
    Q49 → answer not in source — verify against the textbook
50. 50
    Q50 → answer not in source — verify against the textbook
51. 51
    Q51 → answer not in source — verify against the textbook
52. 52
    Q52 → answer not in source — verify against the textbook
53. 53
    Q53 → answer not in source — verify against the textbook
54. 54
    Q54 → answer not in source — verify against the textbook
55. 55
    Q55 → answer not in source — verify against the textbook
56. 56
    Q56 → answer not in source — verify against the textbook
57. 57
    Q57 → answer not in source — verify against the textbook
58. 58
    Q58 → answer not in source — verify against the textbook
59. 59
    Q59 → answer not in source — verify against the textbook
60. 60
    Q60 → answer not in source — verify against the textbook
61. 61
    Q61 → answer not in source — verify against the textbook
62. 62
    Q62 → answer not in source — verify against the textbook
63. 63
    Q63 → answer not in source — verify against the textbook
64. 64
    Q64 → answer not in source — verify against the textbook
65. 65
    Q65 → answer not in source — verify against the textbook
66. 66
    Q66 → answer not in source — verify against the textbook
67. 67
    Q67 → answer not in source — verify against the textbook
68. 68
    Q68 → answer not in source — verify against the textbook
69. 69
    Q69 → answer not in source — verify against the textbook
70. 70
    Q70 → answer not in source — verify against the textbook
71. 71
    Q71 → answer not in source — verify against the textbook
72. 72
    Q72 → answer not in source — verify against the textbook
73. 73
    Q73 → answer not in source — verify against the textbook
74. 74
    Q74 → answer not in source — verify against the textbook
75. 75
    Q75 → answer not in source — verify against the textbook
76. 76
    Q76 → answer not in source — verify against the textbook
77. 77
    Q77 → answer not in source — verify against the textbook
78. 78
    Q78 → answer not in source — verify against the textbook
79. 79
    Q79 → answer not in source — verify against the textbook
80. 80
    Q80 → Option (D) — 1/√(1−y²).

MCQ Questions

1 Mark Questions

1. 1
   Define the order of a differential equation.
2. 2
   State the degree of (d²y/dx²)³ + (dy/dx) = 0. (Ans: 1)
3. 3
   What is the integrating factor of dy/dx + P(x)y = Q(x)? (Ans: e^∫P dx)
4. 4
   How many arbitrary constants appear in the general solution of a first-order equation? (Ans: one)

2 Mark Questions

1. 1
   Find the order and degree. (d²y/dx²)² + (dy/dx) = sin x. (Ans: order 2, degree 2)
2. 2
   Solve dy/dx = x/y by separating variables. (Ans: y²/2 = x²/2 + C, i.e. y² − x² = k)
3. 3
   Find the order and degree of (d³y/dx³) + (dy/dx) = 0.

3 Mark Questions

1. 1
   Solve dy/dx = (x + y)/(x − y), which is homogeneous, using y = vx. (Ans: tan⁻¹(y/x) − ½ ln(x² + y²) = c)
2. 2
   Solve the linear equation dy/dx + 2y = e⁻ˣ with integrating factor e^2x. (Ans: y e^2x = eˣ/3·e^x... verify — y = e⁻ˣ + C e⁻²ˣ)
3. 3
   Verify that y = A cos x + B sin x is a solution of d²y/dx² + y = 0 and identify the arbitrary constants. (Ans: two constants, general solution of order 2)

4 Mark Questions

1. 1
   Solve dy/dx + (y/x) = x³, a linear differential equation, and find the particular solution given y(1) = 1. (Ans: IF = x; y x = x⁵/5 + C; C = 4/5, giving y = x⁴/5 + 4/(5x))
2. 2
   Form the differential equation of the family of circles touching the y-axis at the origin and solve it by the method of homogeneous equations or otherwise. State clearly how the parameter is eliminated.