Differential Equations

1

The order of the differential equation d³y/dx³ + 2(d²y/dx²)² + dy/dx = 0 is:

(A) 1

(B) 2

(C) 3

(D) Not defined

✅ Solution

Answer: (C) 3

The highest order derivative present is d³y/dx³, which is of order 3. Hence the order of the differential equation is 3.

2

The degree of the differential equation (dy/dx)² + sin(dy/dx) = 0 is:

(A) 1

(B) 2

(C) 3

(D) Not defined

✅ Solution

Answer: (D) Not defined

Since the equation contains sin(dy/dx), it is NOT a polynomial equation in its derivatives. Therefore, the degree is not defined.

3

The number of arbitrary constants in the general solution of a 4th order differential equation is:

(A) 0

(B) 2

(C) 3

(D) 4

✅ Solution

Answer: (D) 4

The general solution of an nth order differential equation contains exactly n arbitrary constants. So for order 4, there are 4 arbitrary constants.

4

The general solution of dy/dx = eˣ⁺ʸ is:

(A) eˣ + e⁻ʸ = C

(B) eˣ + eʸ = C

(C) e⁻ˣ + eʸ = C

(D) e⁻ˣ + e⁻ʸ = C

✅ Solution

Answer: (A) eˣ + e⁻ʸ = C

dy/dx = eˣ · eʸ
Separate: e⁻ʸ dy = eˣ dx
Integrate: −e⁻ʸ = eˣ + C₁
Rearranging: eˣ + e⁻ʸ = C

5

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

✅ Solution

Answer: (D) y² dx + (x² − xy − y²)dy = 0

For a homogeneous DE, F(λx, λy) = λⁿ F(x, y). Check option D: dy/dx = −y²/(x² − xy − y²). Replacing x → λx, y → λy gives λ² factor in both numerator and denominator, confirming it is homogeneous of degree zero. Options A has constant terms; B and C are not homogeneous of degree 0.

6

The Integrating Factor (I.F.) of the differential equation x dy/dx − y = 2x² is:

(A) e⁻ˣ

(B) e⁻ʸ

(C) 1/x

(D) x

✅ Solution

Answer: (C) 1/x

Divide by x: dy/dx − (1/x)y = 2x. Here P = −1/x.
I.F = e^∫P dx = e^∫(−1/x)dx = e^(−ln x) = 1/x.

7

The solution of the differential equation dy/dx = (1 + y²)/(1 + x²) is:

(A) tan⁻¹y = tan⁻¹x + C

(B) y = x + C

(C) y² = x² + C

(D) None of these

✅ Solution

Answer: (A) tan⁻¹y = tan⁻¹x + C

Separate variables: dy/(1+y²) = dx/(1+x²)
Integrate both sides: tan⁻¹y = tan⁻¹x + C

8

The number of arbitrary constants in the particular solution of a 3rd order differential equation is:

(A) 3

(B) 2

(C) 1

(D) 0

✅ Solution

Answer: (D) 0

A particular solution is obtained by assigning specific values to all arbitrary constants. Therefore, it contains zero arbitrary constants.

9

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

✅ Solution

Answer: (D) Not defined

Because the equation contains the term sin(dy/dx), it is not a polynomial in its derivatives. The degree is therefore not defined.

10

If Rs 1000 is deposited in a bank at 5% per year (continuous compounding), the amount after 10 years [use e⁰·⁵ ≈ 1.648] is:

(A) Rs 1500

(B) Rs 1648

(C) Rs 1728

(D) Rs 1800

✅ Solution

Answer: (B) Rs 1648

Solving dP/dt = P/20 gives P = 1000 · e^(t/20).
At t = 10: P = 1000 · e^(10/20) = 1000 · e^0.5 = 1000 × 1.648 = Rs 1648.

11

The general solution of x dy/dx + 2y = x² is:

(A) y = x²/4 + Cx⁻²

(B) y = x²/4 + Cx²

(C) y = x²/4 + C

(D) y = x² + C

✅ Solution

Answer: (A) y = x²/4 + Cx⁻²

Divide by x: dy/dx + (2/x)y = x. P = 2/x, Q = x.
I.F = e^∫(2/x)dx = e^(2 ln x) = x².
y · x² = ∫x · x² dx = ∫x³ dx = x⁴/4 + C
∴ y = x²/4 + Cx⁻²

12

The substitution used to solve a homogeneous differential equation dy/dx = F(x, y) where F is of degree 0 is:

(A) y = vx

(B) x = vy

(C) y = v + x

(D) y = v/x

✅ Solution

Answer: (A) y = vx

To solve dy/dx = g(y/x), we substitute y = vx, so dy/dx = v + x(dv/dx). This converts it to a variable-separable form.

13

A bacteria culture has 10⁵ bacteria initially and grows at a rate proportional to the number present. If it increases 10% in 2 hours, in how many hours will the count double?

(A) 2 log₂/log(1.1) hr

(B) 2 log 2/log(1.1) hr

(C) 20 log 2 hr

(D) log 2/log(1.1) hr

✅ Solution

Answer: (B) 2 log 2 / log(1.1) hr

dN/dt = kN → N = N₀ eᵏᵗ
At t = 2: 1.1 N₀ = N₀ e²ᵏ → k = log(1.1)/2
For doubling: 2 N₀ = N₀ eᵏᵗ → t = log2/k = 2 log 2 / log(1.1) hours

14

The general solution of the differential equation e^x dy + (ye^x + 2x) dx = 0 is:

(A) x eʸ + x² = C

(B) x eʸ + y² = C

(C) y eˣ + x² = C

(D) y eʸ + x² = C

✅ Solution

Answer: (C) y eˣ + x² = C

Rewrite: dy/dx + y = −2x e⁻ˣ. P = 1, Q = −2x e⁻ˣ.
I.F = e^∫dx = eˣ.
y · eˣ = ∫(−2x e⁻ˣ · eˣ) dx = ∫−2x dx = −x² + C
∴ y eˣ + x² = C

15

The volume of a spherical balloon being inflated changes at a constant rate. If the radius is initially 3 units and becomes 6 units after 3 seconds, the radius after t seconds is:

(A) (63t + 27)^(1/3)

(B) (63t + 27)^(1/2)

(C) (27t + 63)^(1/3)

(D) (9t + 3)^(1/3)

✅ Solution

Answer: (A) (63t + 27)^(1/3)

dV/dt = k (constant). V = (4/3)πr³.
(4/3)π · 3r² (dr/dt) = k → integrating: (4/3)πr³ = kt + C
At t = 0, r = 3: C = (4/3)π(27) = 36π
At t = 3, r = 6: k·3 + 36π = (4/3)π(216) → k = 84π
∴ r³ = 63t + 27 → r = (63t + 27)^(1/3)

16

Find the order of the differential equation: d⁴y/dx⁴ + sin(y‴) = 0

✅ Solution

Answer: 4

The highest order derivative is d⁴y/dx⁴ (4th order). Hence order = 4.
(Note: degree is not defined because of sin(y‴).)

17

Find the degree of: d³y/dx³ + 2(d²y/dx²)² − dy/dx + y = 0

✅ Solution

Answer: 1

The equation is a polynomial in its derivatives. The highest order derivative is d³y/dx³ and it appears with power 1. So degree = 1.

18

Verify: What is the value of the LHS of d²y/dx² + y = 0 when y = 2sin(x + π/4)? (Write the numerical value)

✅ Solution

Answer: 0

y = 2sin(x + π/4)
dy/dx = 2cos(x + π/4)
d²y/dx² = −2sin(x + π/4) = −y
∴ d²y/dx² + y = −y + y = 0

19

Find the general solution of dy/dx = 4 − y² (−2 < y < 2).
Express the answer in the form sin⁻¹(y/2) = f(x). What is f(x)?

✅ Solution

Answer: f(x) = 2x + C

Separate: dy/√(4 − y²) = dx
∫dy/√(4 − y²) = ∫dx
sin⁻¹(y/2) = x + C₁
∴ sin⁻¹(y/2) = 2x + C where C = 2C₁
So f(x) = 2x + C

20

In a bank, principal increases at 5%/year continuously. In how many years does Rs 1000 double? [Give answer in terms of log.]

✅ Solution

Answer: 20 logₑ 2 years

dP/dt = P/20 → P = 1000 · e^(t/20)
For doubling: 2000 = 1000 · e^(t/20)
t/20 = logₑ 2
∴ t = 20 logₑ 2 ≈ 13.86 years

21

Find the particular solution of dy/dx = −4xy², given y = 1 when x = 0. What is the value of y when x = 1?

✅ Solution

Answer: y = 1/3

Separate: dy/y² = −4x dx
−1/y = −2x² + C
y = 1 at x = 0 → C = −1 → y = 1/(2x² + 1)
At x = 1: y = 1/(2·1 + 1) = 1/3

22

Find the Integrating Factor of: dy/dx + (cot x) · y = 4x cosec x

✅ Solution

Answer: sin x

P = cot x
I.F = e^∫cot x dx = e^(log|sin x|) = sin x

23

For the differential equation (x − y) dy/dx = x + 2y, what substitution makes it variable-separable after confirming it is homogeneous? (State the substitution variable)

✅ Solution

Answer: y = vx

dy/dx = (x + 2y)/(x − y). Put x = λx, y = λy: F(λx, λy) = λ⁰ · F(x,y) → homogeneous degree 0.
Substitution: y = vx, then dy/dx = v + x(dv/dx), leading to a separable equation in v and x.

24

Find the equation of the curve passing through (0, 1) whose slope at any point (x, y) equals x + xy. What is the value of constant C?

✅ Solution

Answer: C = 2

dy/dx − xy = x. Linear DE; P = −x, Q = x.
I.F = e^(−x²/2).
y = −1 + C·e^(x²/2)
At (0,1): 1 = −1 + C·1 → C = 2
Curve: y = −1 + 2e^(x²/2)

25

A rate of change of r% per year causes Rs 100 to double in 10 years. Given logₑ 2 = 0.6931, find r (correct to 2 decimal places).

✅ Solution

Answer: r ≈ 6.93

dP/dt = (r/100)P → P = 100 · e^(rt/100)
At t = 10: 200 = 100 · e^(10r/100) = 100 · e^(r/10)
e^(r/10) = 2 → r/10 = logₑ 2 = 0.6931
∴ r = 6.931 ≈ 6.93

Part A – Multiple Choice Questions

Part B – One-Word / One-Value Answer

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