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Solutions

Overview

Step-by-step NCERT solutions for Differential Equations (Chapter 9, CBSE Class 12 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Differential Equations textbook chapter.

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What these solutions cover

All 98 questions in Differential Equations are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 9.1

  1. d⁴y/dx⁴ + sin(y''') = 0
  2. y' + 5y = 0
  3. (ds/dt)⁴ + 3s * d²s/dt² = 0
  4. (d²y/dx²)² + cos(dy/dx) = 0
  5. d²y/dx² = cos 3x + sin 3x
  6. (y''')² + (y'')³ + (y')⁴ + y⁵ = 0
  7. y''' + 2y'' + y' = 0
  8. y' + y = e^x
  9. y'' + (y')² + 2y = 0
  10. y'' + 2y' + sin y = 0
  11. The degree of the differential equation (d²y/dx²)³ + (dy/dx)² + sin(dy/dx) + 1 = 0 is: (A) 3 (B) 2 (C) 1 (D) not defined
  12. The order of the differential equation 2x² * d²y/dx² - 3 * dy/dx + y = 0 is: (A) 2 (B) 1 (C) 0 (D) not defined

Exercise 9.2

  1. y = e^x + 1 : y'' - y' = 0
  2. y = x² + 2x + C : y' - 2x - 2 = 0
  3. y = cos x + C : y' + sin x = 0
  4. y = sqrt(1 + x²) : y' = xy/(1 + x²)
  5. y = Ax : xy' = y (x ≠ 0)
  6. y = x sin x : xy' = y + x*sqrt(x² - y²) (x ≠ 0, x > y or x < -y)
  7. xy = log y + C : y' = y²/(1 - xy) (xy ≠ 1)
  8. y - cos y = x : (y sin y + cos y + x) y' = y
  9. x + y = tan⁻¹(y) : y² y' + y² + 1 = 0
  10. y = sqrt(a² - x²), x ∈ (-a, a) : x + y * dy/dx = 0 (y ≠ 0)
  11. The number of arbitrary constants in the general solution of a differential equation of fourth order are: (A) 0 (B) 2 (C) 3 (D) 4
  12. The number of arbitrary constants in the particular solution of a differential equation of third order are: (A) 3 (B) 2 (C) 1 (D) 0

Exercise 9.3

  1. dy/dx = (1 - cos x)/(1 + cos x)
  2. dy/dx = sqrt(4 - y²) (-2 < y < 2)
  3. dy/dx + y = 1 (y ≠ 1)
  4. sec² x tan y dx + sec² y tan x dy = 0
  5. (e^x + e⁻x) dy - (e^x - e⁻x) dx = 0
  6. dy/dx = (1 + x²)(1 + y²)
  7. y log y dx - x dy = 0
  8. x⁵ dy/dx = -y⁵
  9. dy/dx = sin⁻¹ x
  10. e^x tan y dx + (1 - e^x) sec² y dy = 0
  11. (x³ + x² + x + 1) dy/dx = 2x² + x; y = 1 when x = 0
  12. x(x² - 1) dy/dx = 1; y = 0 when x = 2
  13. cos(dy/dx) = a (a ∈ R); y = 1 when x = 0
  14. dy/dx = y tan x; y = 1 when x = 0
  15. Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = e^x sin x.
  16. For the differential equation xy dy/dx = (x + 2)(y + 2), find the solution curve passing through the point (1, -1).
  17. Find the equation of a curve passing through the point (0, -2) given that at any point (x, y) on the curve, the product of the slope of its tangent and the y coordinate of the point is equal to the x coordinate of the point.
  18. At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4, -3). Find the equation of the curve given that it passes through (-2, 1).
  19. The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units, find the radius of the balloon after t seconds.
  20. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 doubles itself in 10 years (log_e 2 = 0.6931).
  21. In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e⁰.5 = 1.648).
  22. In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
  23. The general solution of the differential equation dy/dx = e^(x+y) is: (A) e^x + e⁻y = C (B) e^x + e^y = C (C) e⁻x + e^y = C (D) e⁻x + e⁻y = C

Exercise 9.4

  1. (x² + xy) dy = (x² + y²) dx
  2. y' = (x + y)/x
  3. (x - y) dy - (x + y) dx = 0
  4. (x² - y²) dx + 2xy dy = 0
  5. x² dy/dx = x² - 2y² + xy
  6. x dy - y dx = sqrt(x² + y²) dx
  7. {x cos(y/x) + y sin(y/x)} y dx = {y sin(y/x) - x cos(y/x)} x dy
  8. x dy/dx - y + x sin(y/x) = 0
  9. y dx + x log(y/x) dy - 2x dy = 0
  10. (1 + e^(x/y)) dx + e^(x/y)(1 - x/y) dy = 0
  11. (x + y) dy + (x - y) dx = 0; y = 1 when x = 1
  12. x² dy + (xy + y²) dx = 0; y = 1 when x = 1
  13. [x sin²(y/x) - y] dx + x dy = 0; y = pi/4 when x = 1
  14. dy/dx - y/x + cosec(y/x) = 0; y = 0 when x = 1
  15. 2xy + y² - 2x² dy/dx = 0; y = 2 when x = 1
  16. A homogeneous differential equation of the form dx/dy = h(x/y) can be solved by making the substitution: (A) y = vx (B) v = yx (C) x = vy (D) x = v
  17. Which of the following is a homogeneous differential equation? (A) (4x + 6y + 5) dy - (3y + 2x + 4) dx = 0 (B) (xy) dx - (x³ + y³) dy = 0 (C) (x³ + 2y²) dx + 2xy dy = 0 (D) y² dx + (x² - xy - y²) dy = 0

Exercise 9.5

  1. dy/dx + 2y = sin x
  2. dy/dx + 3y = e^(-2x)
  3. dy/dx + y/x = x²
  4. dy/dx + (sec x) y = tan x (0 ≤ x < pi/2)
  5. cos² x * dy/dx + y = tan x (0 ≤ x < pi/2)
  6. x dy/dx + 2y = x² log x
  7. x log x * dy/dx + y = 2/x * log x
  8. (1 + x²) dy + 2xy dx = cot x dx (x ≠ 0)
  9. x dy/dx + y - x + xy cot x = 0 (x ≠ 0)
  10. (x + y) dy/dx = 1
  11. y dx + (x - y²) dy = 0
  12. (x + 3y²) dy/dx = y
  13. dy/dx + 2y tan x = sin x; y = 0 when x = pi/3
  14. (1 + x²) dy/dx + 2xy = 1/(1 + x²); y = 0 when x = 1
  15. dy/dx - 3y cot x = sin 2x; y = 2 when x = pi/2
  16. Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.
  17. Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
  18. The Integrating Factor of the differential equation x dy/dx - y = 2x² is: (A) e^(-x) (B) e^(-y) (C) 1/x (D) x
  19. The Integrating Factor of the differential equation (1 - y²) dx/dy + yx = ay (-1 < y < 1) is: (A) 1/(y²⁻1) (B) 1/sqrt(y²⁻1) (C) 1/(1-y²) (D) 1/sqrt(1-y²)

Miscellaneous Exercise

  1. For each of the differential equations given below, indicate its order and degree (if defined).
    • (i) d²y/dx² + 5x(dy/dx)² - 6y = log x
    • (ii) (dy/dx)³ - 4(dy/dx)² + 7y = sin x
    • (iii) d⁴y/dx⁴ - sin(d³y/dx³) = 0
  2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
    • (i) xy = a e^x + b e⁻x + x² : x d²y/dx² + 2 dy/dx - xy + x² - 2 = 0
    • (ii) y = e^x(a cos x + b sin x) : d²y/dx² - 2 dy/dx + 2y = 0
    • (iii) y = x sin 3x : d²y/dx² + 9y - 6 cos 3x = 0
    • (iv) x² = 2y² log y : (x² + y²) dy/dx - xy = 0
  3. Prove that x² - y² = c(x² + y²)² is the general solution of differential equation (x³ - 3xy²) dx = (y³ - 3x² y) dy, where c is a parameter.
  4. Find the general solution of the differential equation dy/dx + sqrt((1 - y²)/(1 - x²)) = 0.
  5. Show that the general solution of the differential equation dy/dx + (y² + y + 1)/(x² + x + 1) = 0 is given by (x + y + 1) = A(1 - x - y - 2xy), where A is parameter.
  6. Find the equation of the curve passing through the point (0, pi/4) whose differential equation is sin x cos y dx + cos x sin y dy = 0.
  7. Find the particular solution of the differential equation (1 + e^(2x)) dy + (1 + y²) e^x dx = 0, given that y = 1 when x = 0.
  8. Solve the differential equation y e^(x/y) dx = (x e^(x/y) + y²) dy (y ≠ 0).
  9. Find a particular solution of the differential equation (x - y)(dx + dy) = dx - dy, given that y = -1 when x = 0. (Hint: put x - y = t)
  10. Solve the differential equation [e^(-2*sqrt(x))/sqrt(x) - y/sqrt(x)] dx/dy = 1 (x ≠ 0).
  11. Find a particular solution of the differential equation dy/dx + y cot x = 4x cosec x (x ≠ 0), given that y = 0 when x = pi/2.
  12. Find a particular solution of the differential equation (x + 1) dy/dx = 2 e⁻y - 1, given that y = 0 when x = 0.
  13. The general solution of the differential equation (y dx - x dy)/y = 0 is: (A) xy = C (B) x = Cy² (C) y = Cx (D) y = Cx²
  14. The general solution of a differential equation of the type dx/dy + P_1 x = Q_1 is: (A) y e^(∫ P_1 dy) = ∫(Q_1 e^(∫ P_1 dy)) dy + C (B) y e^(∫ P_1 dx) = ∫(Q_1 e^(∫ P_1 dx)) dx + C (C) x e^(∫ P_1 dy) = ∫(Q_1 e^(∫ P_1 dy)) dy + C (D) x e^(∫ P_1 dx) = ∫(Q_1 e^(∫ P_1 dx)) dx + C
  15. The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is: (A) x e^y + x² = C (B) x e^y + y² = C (C) y e^x + x² = C (D) y e^y + x² = C
Sample

Worked examples

3 questions from Differential Equations, solved step by step.

Exercise 9.1

d⁴y/dx⁴ + sin(y''') = 0

The highest order derivative present is d⁴y/dx⁴, so the order is 4.

Since the equation contains sin(y'''), it is NOT a polynomial equation in its derivatives.

Therefore, the degree is not defined.

Answer: Order: 4, Degree: not defined

Exercise 9.2

y = e^x + 1 : y'' - y' = 0

Given y = e^x + 1.

Differentiate: y' = e^x.

Differentiate again: y'' = e^x.

Now y'' - y' = e^x - e^x = 0 = R.H.S.

Hence y = e^x + 1 is a solution of y'' - y' = 0.

Answer: Verified: y = e^x + 1 satisfies y'' - y' = 0.

Exercise 9.3

dy/dx = (1 - cos x)/(1 + cos x)

Use the identities: 1 - cos x = 2 sin²(x/2) and 1 + cos x = 2 cos²(x/2).

So dy/dx = 2 sin²(x/2) / (2 cos²(x/2)) = tan²(x/2) = sec²(x/2) - 1.

Separate variables: dy = (sec²(x/2) - 1) dx.

Integrate both sides:

y = 2 tan(x/2) - x + C.

Answer: y = 2 tan(x/2) - x + C

All 98 questions in Differential Equations are solved step by step in the Solutions PDF above.

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