68 free MCQs on Differential Equations with worked answers and explanations. Equations involving derivatives and their solutions
Take the timed Differential Equations chapterwise test →Below are 68 practice questions on Differential Equations, sorted Easy → Hard. Tap “Show answer & explanation” under any question to check yourself. Want the full theory first? Read the Differential Equations notes.
The general solution y=Ax² represents an entire FAMILY of curves, one for each value of the arbitrary constant A; a particular solution (fixed by an initial condition) selects exactly one curve from this family.
Q1.
A differential equation involves:
Answer: B. Derivatives of a function
Why: A differential equation contains derivatives (dy/dx, d²y/dx², etc.) of one or more functions.
Q2.
The order of a differential equation is the order of its:
Answer: B. Highest derivative
Why: Order = order of the highest derivative present. For d²y/dx² + dy/dx = 0, the order is 2.
Q3.
The degree of a differential equation is:
Answer: B. Power of highest-order derivative (when rationalized)
Why: Degree = power of the highest-order derivative after clearing radicals and fractions.
Q4.
Order of: d²y/dx² + 3(dy/dx) + 2y = 0
Answer: B. 2
Why: Highest derivative is d²y/dx² (second derivative). Order = 2.
Q5.
The degree of (dy/dx)³ + y = 0 is:
Answer: C. 3
Why: Highest derivative is dy/dx with power 3. Degree = 3.
Q6.
Solve dy/dx = x. The general solution is:
Answer: B. y = x²/2 + C
Why: Integrate both sides: y = x²/2 + C.
Q7.
Solve dy/dx = y. The solution is:
Answer: B. y = Ce<sup>x</sup>
Why: Separate: dy/y = dx. ln|y| = x + const. y = Ce<sup>x.</sup>
Q8.
A general solution of a first-order DE contains:
Answer: B. 1 arbitrary constant
Why: First-order DE general solution has 1 arbitrary constant (from one integration).
Q9.
A particular solution is obtained by:
Answer: B. Applying initial/boundary conditions to determine the constant
Why: Particular solution: substitute given initial conditions into general solution to find the specific constant value.
Q10.
A differential equation of order 2 has how many arbitrary constants in general solution?
Answer: C. 2
Why: An nth order DE has n arbitrary constants in the general solution.
Q11.
Separation of variables method works when:
Answer: B. All terms with y (and dy) can be separated from terms with x (and dx)
Why: Separation of variables: rewrite as f(y)dy = g(x)dx and integrate both sides separately.
Q12.
The integrating factor for dy/dx + Py = Q (P, Q functions of x) is:
Answer: B. e<sup>integral of P dx</sup>
Why: Linear first-order DE: integrating factor = e<sup>integral P dx</sup>. Multiply both sides to make LHS exact.
Q13.
Solve dy/dx = k (constant). Solution:
Answer: B. y = kx + C
Why: Integrate: y = kx + C. Straight line with slope k.
Q14.
Which of these is a first-order linear DE?
Answer: B. dy/dx + y = x (linear in y)
Why: dy/dx + P(x)y = Q(x) is linear first-order. y² or (dy/dx)² makes it nonlinear.
Q15.
A homogeneous DE of the form dy/dx = f(y/x) is solved by substituting:
Answer: A. y = vx
Why: Substitute y = vx (so dy/dx = v + x dv/dx). Transforms homogeneous DE to separable form.
Q16.
The differential equation of all circles with centre at origin:
Answer: A. x + y dy/dx = 0
Why: x² + y² = r². Differentiate: 2x + 2y(dy/dx) = 0, so x + y(dy/dx) = 0.
Q17.
Exact DE: M dx + N dy = 0 is exact when:
Answer: C. dM/dy = dN/dx
Why: Exactness condition: partial derivative of M with respect to y = partial derivative of N with respect to x.
Q18.
The complementary function (CF) of a linear DE with constant coefficients is the:
Answer: B. Solution when right-hand side = 0 (homogeneous part)
Why: CF: general solution of the homogeneous equation (RHS = 0). The complete solution = CF + PI (particular integral).
Q19.
For exponential growth model: dP/dt = kP (k > 0), the solution is:
Answer: B. P = P₀ e<sup>kt</sup>
Why: Separate: dP/P = k dt. Integrate: ln P = kt + C. P = P₀ e<sup>kt</sup>. Classic exponential growth.
Q20.
A differential equation is linear if:
Answer: A. The dependent variable and all its derivatives appear with power 1
Why: Linear DE: dependent variable (y) and all its derivatives appear with power 1. No products like y × y', etc.
Q21.
Solve dy/dx = x/y. The general solution:
Answer: C. x² + y² = C
Why: y dy = x dx. Integrate: y²/2 = x²/2 + C/2. So x² - y² = constant, or x² + y² = C if we separate differently. Actually: y dy = x dx gives y²/2 = x²/2 + k, so y² - x² = C.
Q22.
Solve the linear DE: dy/dx + y = eˣ.
Answer: B. y = eˣ/2 + Ce<sup>-x</sup>
Why: IF = e<sup>integral 1 dx</sup> = eˣ. Multiply: (yeˣ)' = e²ˣ. yeˣ = e²ˣ/2 + C. y = eˣ/2 + Ce<sup>-x</sup>.
Q23.
The DE dy/dx = (x + y)/(x - y) is:
Answer: B. Homogeneous (degree 1)
Why: f(tx, ty) = (tx+ty)/(tx-ty) = t(x+y)/t(x-y) = f(x,y). Degree 1 homogeneous. Substitute y = vx.
Q24.
The general solution of d²y/dx² + y = 0 (auxiliary equation m² + 1 = 0):
Answer: A. y = A sinx + B cosx
Why: Auxiliary: m² = -1, m = ±i. CF for complex roots ±αi: y = A cos(αx) + B sin(αx). Here α=1.
Q25.
Auxiliary equation for d²y/dx² - 5dy/dx + 6y = 0 is:
Answer: A. m² - 5m + 6 = 0
Why: Replace d<sup>n</sup> y/dx<sup>n</sup> with m<sup>n</sup>: m² - 5m + 6 = 0. Roots: (m-2)(m-3) = 0, m = 2, 3.
Q26.
For d²y/dx² - 5dy/dx + 6y = 0 with roots m = 2, 3, the general solution:
Answer: A. y = Ae²ˣ + Be³ˣ
Why: Two distinct real roots m₁=2, m₂=3: y = Ae<sup>m₁x</sup> + Be<sup>m₂x</sup> = Ae²ˣ + Be³ˣ.
Q27.
If auxiliary equation has repeated root m = 2 (twice), CF is:
Answer: B. y = (A+Bx)e²ˣ
Why: Repeated root m: CF = (A + Bx)e<sup>mx</sup>. For m=2: y = (A+Bx)e²ˣ.
Q28.
Solve dy/dx = (x² + y²)/(2xy) (Bernoulli-type homogeneous):
Answer: B. y² - x² = Cx
Why: Homogeneous: let y=vx. After substitution and simplification: y² - x² = Cx.
Q29.
Population growth modeled as dP/dt = kP, with P(0) = 1000, P(1) = 1500. Find k:
Answer: A. ln(1.5)
Why: P = 1000 e<sup>kt</sup>. At t=1: 1500 = 1000 e<sup>k.</sup> k = ln(1.5).
Q30.
A 2nd order linear DE is said to be homogeneous if:
Answer: B. The right-hand side is zero
Why: Homogeneous DE: the forcing function (RHS) is zero. f(x,y,y',y'') = 0 form with no external forcing.
Q31.
The Wronskian of two solutions y₁ and y₂ of a 2nd order linear homogeneous DE is W = y₁y₂' - y₁'y₂. If W ≠ 0, the solutions are:
Answer: C. Linearly independent
Why: Wronskian test: W ≠ 0 implies y₁ and y₂ are linearly independent (form a fundamental set of solutions).
Q32.
The particular integral (PI) for a non-homogeneous DE is found by:
Answer: B. Method of undetermined coefficients or variation of parameters
Why: PI: particular solution of the non-homogeneous DE. Found by undetermined coefficients (for standard RHS) or variation of parameters.
Q33.
Newton law of cooling: dT/dt = -k(T - T<sub>env</sub>). General solution:
Answer: A. T = T<sub>env</sub> + Ce<sup>-kt</sup>
Why: Let u = T - T<sub>env</sub>. du/dt = -ku. u = Ce<sup>-kt</sup>. T = T<sub>env</sub> + Ce<sup>-kt</sup>.
Q34.
The equation dy/dx + P(x)y = Q(x)y<sup>n</sup> (n not= 0,1) is called:
Answer: B. Bernoulli DE
Why: Bernoulli equation: dy/dx + P(x)y = Q(x)y<sup>n.</sup> Solved by substituting z = y<sup>1-n</sup>.
Q35.
Solve (2x + y) dx + (x + 2y) dy = 0 (check exactness):
Answer: A. x² + xy + y² = C
Why: M=2x+y, N=x+2y. dM/dy=1=dN/dx=1. Exact. Integral: x²+xy+y²=C.
Q36.
The condition for M dx + N dy = 0 to be exact is:
Answer: B. dM/dy = dN/dx
Why: Exactness: partial M/partial y = partial N/partial x (equality of mixed partials).
Q37.
Logistic growth model: dP/dt = rP(1 - P/K). The carrying capacity is:
Answer: B. K
Why: Logistic equation: K is the carrying capacity (maximum sustainable population). r is growth rate.
Q38.
d²y/dx² = 0 has solution:
Answer: B. y = ax + b (linear)
Why: Integrate twice: dy/dx = A (constant). y = Ax + B. Linear function.
Q39.
Method of variation of parameters applies to:
Answer: B. Non-homogeneous linear DE when particular integral cannot be guessed
Why: Variation of parameters: powerful method for finding particular integral of non-homogeneous linear DE, especially when RHS has non-standard forms.
Q40.
The complementary function of (D² - 3D + 2)y = e<sup>3x</sup> (D = d/dx) involves:
Answer: A. e<sup>x</sup> and e<sup>2x</sup>
Why: Auxiliary: m² - 3m + 2 = 0, (m-1)(m-2) = 0. m = 1, 2. CF = Ae<sup>x</sup> + Be<sup>2x</sup>.
Q41.
Solve: y'' - 4y = 0 with y(0) = 1, y'(0) = 0.
Answer: A. y = cosh(2x)
Why: Auxiliary: m² = 4, m = ±2. y = Ae<sup>2x</sup> + Be<sup>-2x</sup>. y(0)=A+B=1, y'(0)=2A-2B=0 so A=B=1/2. y = (e<sup>2x</sup>+e<sup>-2x</sup>)/2 = cosh(2x).
Q42.
The Fourier series of a periodic function uses:
Answer: B. Sinusoids and cosinusoids (trigonometric basis)
Why: A periodic function f(x) with period 2L is expressed as f(x)=a₀/2+Σ[aₙcos(nπx/L)+bₙsin(nπx/L)]. Coefficients found by orthogonality of sin/cos. This is an infinite sum of harmonically related sinusoids and cosinusoids.
Q43.
The method of undetermined coefficients for PI: if f(x) = xe<sup>2x</sup>, the trial PI is:
Answer: B. (Ax + B)e<sup>2x</sup>
Why: Rule: for f(x)=pₙ(x)eᵃˣ where pₙ is degree-n polynomial, trial PI=(Ax+B)e<sup>2x</sup> (same degree polynomial times e<sup>2x</sup>), provided 2 is not a root of the characteristic equation. Substitute into DE to find A and B.
Q44.
The DE x dy/dx - y = x² is solvable by dividing through by x to give:
Answer: A. dy/dx - y/x = x (linear in y)
Why: Divide by x: dy/dx − y/x = x. Linear first-order with P(x)=−1/x, Q(x)=x. Integrating factor: e<sup>∫−1/x dx</sup>=e<sup>−lnx</sup>=1/x. Multiply: d(y/x)/dx=1. Integrate: y/x=x+C, so y=x²+Cx.
Q45.
The general solution of a non-homogeneous linear DE is:
Answer: C. CF + PI (complementary function + particular integral)
Why: Complete solution = CF (solution of homogeneous part, with n constants) + PI (one particular solution of non-homogeneous part).
Q46.
The Bessel equation is:
Answer: B. x²d<sup>2</sup>y/dx<sup>2</sup> + xy_prime + (x²-n²)y = 0
Why: Bessel's equation: x²y''+xy'+(x²−n²)y=0, where n is the order. Solutions are Bessel functions of first kind Jₙ(x) and second kind Yₙ(x), expressed as power series. It arises in problems with cylindrical symmetry (wave equation in cylinders).
Q47.
Euler-Cauchy equation x<sup>n</sup> y<sup>n</sup> + ... has solutions of the form:
Answer: B. x<sup>m</sup> (power function)
Why: For Euler-Cauchy equation x²y''+bxy'+cy=0, substitute y=xᵐ. Then y'=mxᵐ⁻¹, y''=m(m−1)xᵐ⁻². Equation reduces to m(m−1)+bm+c=0 (indicial equation). Roots m determine the power-function solutions xᵐ.
Q48.
d²y/dx² + 4y = cos 2x. The particular integral contains resonance because:
Answer: B. cos 2x frequency matches the homogeneous solution frequency (2)
Why: Resonance: forcing frequency equals natural frequency. Homogeneous solution has cos 2x, same as forcing. PI requires extra factor x: y<sub>P</sub> = x(A cos 2x + B sin 2x).
Q49.
The existence and uniqueness theorem for y' = f(x,y) with y(x₀) = y₀ requires f to be:
Answer: B. Continuous and Lipschitz in y (near the initial point)
Why: Picard-Lindelof theorem: unique solution exists if f is continuous and satisfies Lipschitz condition in y in a rectangle around (x₀,y₀).
Q50.
Solve: dy/dx = -y/x with y(1) = 2.
Answer: A. y = 2/x
Why: Separating variables: dy/y=−dx/x. Integrating both sides: ln|y|=−ln|x|+C₁, so y=A/x. Applying IC y(1)=2: A=2. Solution: y=2/x (equivalently xy=2).
Q51.
The Green function approach solves:
Answer: B. Linear non-homogeneous DEs with arbitrary forcing using superposition
Why: Green function: constructs solution to linear DE with arbitrary forcing. G(x,s) is response to unit impulse at s; full solution = integral of G × f.
Q52.
The order of the differential equation d²y/dx² + 3(dy/dx) + 2y = 0 is:
Answer: A. 2
Why: Order is the highest derivative present, here d²y/dx², so the order is 2.
Q53.
The degree of the differential equation (dy/dx)² + y = 0 is:
Answer: A. 2
Why: Degree is the power of the highest-order derivative once it is a polynomial; here (dy/dx)² gives degree 2.
Q54.
The order of the differential equation dy/dx = x + y is:
Answer: A. 1
Why: The highest derivative is the first derivative dy/dx, so the order is 1.
Q55.
The general solution of the differential equation dy/dx = 0 is:
Answer: A. y = C
Why: If the derivative is zero everywhere, y is a constant, so y = C.
Q56.
The solution of the differential equation dy/dx = ky is:
Answer: A. y = C·e<sup>kx</sup>
Why: Separating variables gives dy/y = k dx, so ln y = kx + c and y = C e<sup>kx</sup>.
Q57.
The number of arbitrary constants in the general solution of a second-order differential equation is:
Answer: A. 2
Why: The general solution of an nth-order equation contains n arbitrary constants, so a second-order one has 2.
Q58.
An equation of the form dy/dx + Py = Q (P, Q functions of x) is called a ___ differential equation:
Answer: A. linear
Why: This standard form is a first-order linear differential equation.
Q59.
The integrating factor of the differential equation dy/dx + y = x is:
Answer: A. e<sup>x</sup>
Why: IF = e<sup>∫P dx</sup> = e<sup>∫1 dx</sup> = e<sup>x.</sup>
Q60.
The differential equation of the family of lines y = mx (m arbitrary) is:
Answer: A. dy/dx = y/x
Why: From y = mx, m = y/x, and dy/dx = m, so dy/dx = y/x.
Q61.
The general solution of dy/dx = y/x is:
Answer: B. y = Cx
Why: Separating variables gives ln y = ln x + c, so y = Cx.
Q62.
The order and degree of (d²y/dx²)³ + (dy/dx)² + y = 0 are:
Answer: B. order 2, degree 3
Why: The highest derivative is second order, raised to the power 3, so order 2 and degree 3.
Q63.
The integrating factor of dy/dx + 2y = e<sup>x</sup> is:
Answer: B. e<sup>2x</sup>
Why: IF = e<sup>∫2 dx</sup> = e<sup>2x</sup>.
Q64.
The general solution of dy/dx = e<sup>x − y</sup> is:
Answer: A. e<sup>y</sup> = e<sup>x</sup> + C
Why: Separating: e<sup>y</sup> dy = e<sup>x</sup> dx, integrating gives e<sup>y</sup> = e<sup>x</sup> + C.
Q65.
The general solution of dy/dx + y·tanx = secx is:
Answer: B. y = sinx + C cosx
Why: IF = secx; (y secx)' = sec²x, so y secx = tanx + C, giving y = sinx + C cosx.
Q66.
The differential equation whose general solution is y = A cosx + B sinx is:
Answer: B. y'' + y = 0
Why: Differentiating twice gives y'' = −y, i.e. y'' + y = 0.
Q67.
The general solution of x(dy/dx) + y = x² is:
Answer: A. y = x²/3 + C/x
Why: IF = x, so (xy)' = x²; integrating xy = x³/3 + C, hence y = x²/3 + C/x.
Q68.
The degree of the differential equation (1 + (dy/dx)²)<sup>3/2</sup> = k·(d²y/dx²) is:
Answer: B. 2
Why: Squaring to clear the fractional power gives (1 + (y')²)³ = k²(y'')², so the degree in the highest derivative is 2.