Differential Equations — Class 12 MCQs with Answers

Differential Equations — important questions & MCQs with answers (Class 12 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   Order of d²y/dx² + dy/dx + y = 0:

   A.3

   B.1

   C.2✓ Correct

   D.0

   Answer: C. 2

   Explanation: Order is the highest derivative present, not the number of terms. Here d²y/dx² appears, so order = 2 — don't stop at dy/dx and call it order 1.

2. Q2Easy

   dy/dx = y type of equation:

   A.Linear only

   B.Variable separable✓ Correct

   C.Bernoulli

   D.Exact

   Answer: B. Variable separable

   Explanation: dy/dx = y separates cleanly into dy/y = dx, with only y-terms on one side and only dx on the other — that split is the defining test for variable separable.

3. Q3Easy

   A differential equation is:

   A.Only involves integers

   B.Algebraic equation

   C.An equation containing derivatives of dependent variable✓ Correct

   D.Linear function

   Answer: C. An equation containing derivatives of dependent variable

   Explanation: A differential equation must contain derivatives of a dependent variable with respect to an independent one — a plain algebraic equation with no derivative term doesn't qualify.

4. Q4Easy

   A solution of DE is:

   A.Random function

   B.A function y = f(x) that satisfies the equation✓ Correct

   C.Always polynomial

   D.Always trigonometric

   Answer: B. A function y = f(x) that satisfies the equation

   Explanation: A solution of a DE is any function y = f(x) that, when substituted back with its derivatives, satisfies the equation identically — it need not be polynomial or trigonometric.

5. Q5Easy

   Variable separable DE has form:

   A.f(x, y) = 0

   B.y = f(x)

   C.dy/dx = constant

   D.f(x) dx = g(y) dy✓ Correct

   Answer: D. f(x) dx = g(y) dy

   Explanation: Variable separable means every x-term (with dx) can be gathered on one side and every y-term (with dy) on the other: f(x) dx = g(y) dy, ready to integrate independently.

6. Q6Easy

   Linear DE of 1st order has form:

   A.dy = dx

   B.dy/dx = y

   C.y' = constant

   D.dy/dx + P(x) y = Q(x)✓ Correct

   Answer: D. dy/dx + P(x) y = Q(x)

   Explanation: The standard first-order linear form is dy/dx + P(x)y = Q(x), where both P and Q can depend on x — dy/dx = y is just the special case P = −1, Q = 0.

7. Q7Medium

   Degree of (dy/dx)² + 3 dy/dx + y = 0:

   A.3

   B.1

   C.2✓ Correct

   D.0

   Answer: C. 2

   Explanation: Degree is the power of the highest derivative once the equation is polynomial in derivatives. dy/dx is squared here, so degree = 2 — don't confuse it with the order, which is 1.

8. Q8Medium

   Linear DE dy/dx + Py = Q has integrating factor:

   A.e^x

   B.e^(∫P dx)✓ Correct

   C.P·Q

   D.Q/P

   Answer: B. e^(∫P dx)

   Explanation: For dy/dx + Py = Q, multiplying by e^(∫P dx) turns the left side into the exact derivative d(y·IF)/dx, letting you integrate directly — plain eˣ only works if P = 1.

9. Q9Medium

   DE for exponential growth: dy/dt = ky (k > 0). Solution:

   A.y = 1/y₀

   B.y = kt

   C.y = e^t

   D.y = y₀ e^(kt)✓ Correct

   Answer: D. y = y₀ e^(kt)

   Explanation: dy/dt = ky separates to dy/y = k dt; integrating from the initial value y₀ gives y = y₀e^(kt) — the constant k sits in the exponent, driving growth when k > 0.

10. Q10Medium

    Solve dy/dx = (1 + y²)/(1 + x²):

    A.tan⁻¹ y = tan⁻¹ x + C✓ Correct

    B.y = x + C

    C.y = (1 + x²)/(1 + y²)

    D.y = arctan x

    Answer: A. tan⁻¹ y = tan⁻¹ x + C

    Explanation: Separating gives dy/(1+y²) = dx/(1+x²), and both sides integrate to inverse tangents: tan⁻¹y = tan⁻¹x + C — don't mistake this for a linear relation like y = x + C.

11. Q11Medium

    Solve dy/dx + y = eˣ:

    A.y = e^(2x)

    B.y = eˣ

    C.y = (eˣ + C)/eˣ · ... actually y = (e^(2x))/2 · e^(−x) + C e^(−x)✓ Correct

    D.Cannot solve

    Answer: C. y = (eˣ + C)/eˣ · ... actually y = (e^(2x))/2 · e^(−x) + C e^(−x)

    Explanation: IF = eˣ turns the left side into d(eˣy)/dx = e^(2x); integrating gives eˣy = e^(2x)/2 + C, so y = eˣ/2 + Ce^(−x) — always divide back by the IF at the end.

12. Q12Hard

    A homogeneous DE is solved by substitution:

    A.y = 0

    B.y = e^x

    C.y = sin x

    D.y = vx✓ Correct

    Answer: D. y = vx

    Explanation: A homogeneous DE has dy/dx depending only on the ratio y/x, so substituting y = vx (dy/dx = v + x·dv/dx) turns it into a separable equation in v and x.

13. Q13Hard

    Find DE whose general solution is y = A cos x + B sin x:

    A.d²y/dx² − y = 0

    B.dy/dx + y = 0

    C.d²y/dx² + y = 0✓ Correct

    D.dy/dx − y = 0

    Answer: C. d²y/dx² + y = 0

    Explanation: Differentiating y = A cos x + B sin x twice gives y'' = −A cos x − B sin x = −y, so d²y/dx² + y = 0 — this eliminates both A and B at once.

14. Q14Hard

    Solve dy/dx = xy / (1 + x²):

    A.y = ln(1 + x²)

    B.y = 1 + x

    C.y = √(1 + x²)

    D.y² = C(1 + x²)✓ Correct

    Answer: D. y² = C(1 + x²)

    Explanation: Separating gives dy/y = x/(1+x²) dx; integrating gives ln y = (1/2)ln(1+x²) + C, so squaring clears the half-power to give y² = C(1+x²).

Differential Equations — FAQs

More Class 12 Mathematics chapters

• 01Relations and Functions
• 02Inverse Trigonometric Functions
• 03Matrices
• 04Determinants
• 05Continuity and Differentiability
• 06Application of Derivatives

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