68 practice questions on Differential Equations , sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Differential Equations notes .
Family of Solution Curves: y = Ax² Each value of the constant A gives ONE particular curve from the family 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.
Easy - 20 questions Q1.
A differential equation involves:
A Algebraic variables without any calculus operationsB Derivatives of a functionC Integral expressions rather than derivative onesD Constant terms with no variable componentShow answer & explanation →
Q2.
The order of a differential equation is the order of its:
A Highest powerB Highest derivativeC Number of variablesD Number of constantsShow answer & explanation →
Q3.
The degree of a differential equation is:
A The order of the highest derivative appearing in the equationB Power of highest-order derivative (when rationalized)C The total number of independent solutions the equation admitsD The total number of additive terms in the equationShow answer & explanation →
Q8.
A general solution of a first-order DE contains:
A No constantsB 1 arbitrary constantC 2 arbitrary constantsD Only x termsShow answer & explanation →
Q9.
A particular solution is obtained by:
A Differentiating the general solution with respect to xB Applying initial/boundary conditions to determine the constantC Multiplying the general solution by an integrating factorD Determining the order of the highest derivative presentShow answer & explanation →
Q11.
Separation of variables method works when:
A The equation is linear in y with constant coefficients throughoutB All terms with y (and dy) can be separated from terms with x (and dx)C The equation is generally nonlinear in both the x and y terms togetherD The order of the differential equation exceeds two in this caseShow answer & explanation →
Q12.
The integrating factor for dy/dx + Py = Q (P, Q functions of x) is:
A e<sup>integral of Q dx</sup>B e<sup>integral of P dx</sup>C P × QD 1/PShow answer & explanation →
Q14.
Which of these is a first-order linear DE?
A dy/dx = y², which is nonlinear due to the squared y termB dy/dx + y = x (linear in y)C (dy/dx)² = x, nonlinear because the derivative is squaredD d²y/dx² = y, which is second order, not first orderShow answer & explanation →
Q15.
A homogeneous DE of the form dy/dx = f(y/x) is solved by substituting:
A y = vxB y = x + vC v = x/yD y = x²Show answer & explanation →
Q16.
The differential equation of all circles with centre at origin:
A x + y dy/dx = 0B x dy/dx - y = 0C x dy/dx + y = 0D x² + y² = r²Show answer & explanation →
Q18.
The complementary function (CF) of a linear DE with constant coefficients is the:
A Solution obtained directly from the right-hand side forcing termB Solution when right-hand side = 0 (homogeneous part)C The antiderivative of the entire differential equationD The multiplying factor that converts the equation into exact formShow answer & explanation →
Q19.
For exponential growth model: dP/dt = kP (k > 0), the solution is:
A P = P₀ + ktB P = P₀ e<sup>kt</sup>C P = ktD P = P₀/kShow answer & explanation →
Q20.
A differential equation is linear if:
A The dependent variable and all its derivatives appear with power 1B First-order derivatives specifically are present, regardless of their powersC The independent variable x appears, with y largely absent from the expressionD All the coefficients multiplying the derivatives are constantsShow answer & explanation →
Medium - 20 questions Q22.
Solve the linear DE: dy/dx + y = eˣ.
A y = (x+C)e<sup>-x</sup>B y = eˣ/2 + Ce<sup>-x</sup>C y = eˣ + CD y = Ce<sup>x</sup>Show answer & explanation →
Q23.
The DE dy/dx = (x + y)/(x - y) is:
A Exact, requiring a specific test conditionB Homogeneous (degree 1)C Linear, requiring a different solving methodD Separable, allowing direct variable splittingShow answer & explanation →
Q24.
The general solution of d²y/dx² + y = 0 (auxiliary equation m² + 1 = 0):
A y = A sinx + B cosxB y = Ae<sup>x</sup> + Be<sup>-x</sup>C y = (A+Bx)e<sup>x</sup>D y = Ae<sup>ix</sup>Show answer & explanation →
Q25.
Auxiliary equation for d²y/dx² - 5dy/dx + 6y = 0 is:
A m² - 5m + 6 = 0B m² + 5m + 6 = 0C m² - 5m - 6 = 0D 5m² - m + 6 = 0Show answer & explanation →
Q26.
For d²y/dx² - 5dy/dx + 6y = 0 with roots m = 2, 3, the general solution:
A y = Ae²ˣ + Be³ˣB y = (A+Bx)e²ˣC y = A sin 2x + B cos 3xD y = Ae<sup>x</sup> + Be<sup>6x</sup>Show answer & explanation →
Q27.
If auxiliary equation has repeated root m = 2 (twice), CF is:
A y = Ae²ˣB y = (A+Bx)e²ˣC y = A cos 2x + B sin 2xD y = Ae²ˣ + Be<sup>-2x</sup>Show answer & explanation →
Q28.
Solve dy/dx = (x² + y²)/(2xy) (Bernoulli-type homogeneous):
A x² - y² = CxB y² - x² = CxC y = Cx²D x² + y² = CxShow answer & explanation →
Q29.
Population growth modeled as dP/dt = kP, with P(0) = 1000, P(1) = 1500. Find k:
A ln(1.5)B 0.5C ln(2)D 1.5Show answer & explanation →
Q30.
A 2nd order linear DE is said to be homogeneous if:
A All coefficients are constantB The right-hand side is zeroC It has constant coefficients and zero RHSD Degree is 1Show answer & explanation →
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:
A IdenticalB Linearly dependentC Linearly independentD ImaginaryShow answer & explanation →
Q32.
The particular integral (PI) for a non-homogeneous DE is found by:
A Solving the homogeneous part and treating it as the full solutionB Method of undetermined coefficients or variation of parametersC Factoring the differential equation into linear operator piecesD Finding the roots of the auxiliary equation aloneShow answer & explanation →
Q33.
Newton law of cooling: dT/dt = -k(T - T<sub>env</sub>). General solution:
A T = T<sub>env</sub> + Ce<sup>-kt</sup>B T = Ce<sup>kt</sup>C T = T<sub>env</sub> - ktD T = T<sub>env</sub> + ktShow answer & explanation →
Q34.
The equation dy/dx + P(x)y = Q(x)y<sup>n</sup> (n not= 0,1) is called:
A Linear DEB Bernoulli DEC Exact DED Homogeneous DEShow answer & explanation →
Q35.
Solve (2x + y) dx + (x + 2y) dy = 0 (check exactness):
A x² + xy + y² = CB x² + 2xy + y² = CC x² + xy = CD Not exactShow answer & explanation →
Q39.
Method of variation of parameters applies to:
A First-order equations specifically, regardless of homogeneity statusB Non-homogeneous linear DE when particular integral cannot be guessedC Exact equations specifically, where an integrating factor already existsD Separable equations where the variables split cleanly apartShow answer & explanation →
Q40.
The complementary function of (D² - 3D + 2)y = e<sup>3x</sup> (D = d/dx) involves:
A e<sup>x</sup> and e<sup>2x</sup>B e<sup>3x</sup>C sin x and cos xD e<sup>-x</sup> and e<sup>-2x</sup>Show answer & explanation →
Hard - 28 questions Q41.
Solve: y'' - 4y = 0 with y(0) = 1, y'(0) = 0.
A y = cosh(2x)B y = e<sup>2x</sup> + e<sup>-2x</sup>C y = cos(2x)D y = sinh(2x)Show answer & explanation →
Q42.
The Fourier series of a periodic function uses:
A Taylor polynomials expanded about a single fixed pointB Sinusoids and cosinusoids (trigonometric basis)C Polynomial terms mainly, with few trigonometric componentsD Real exponential terms mainly, with little oscillatory behaviorShow answer & explanation →
Q43.
The method of undetermined coefficients for PI: if f(x) = xe<sup>2x</sup>, the trial PI is:
A Axe<sup>2x</sup>B (Ax + B)e<sup>2x</sup>C Ae<sup>2x</sup>D (Ax² + Bx)e<sup>2x</sup>Show answer & explanation →
Q44.
The DE x dy/dx - y = x² is solvable by dividing through by x to give:
A dy/dx - y/x = x (linear in y)B y dy/dx = x, an unrelated separable rearrangementC dy/dx = x², ignoring the y term on the left sideD dy/dx + y/x = x, with the sign on y/x flippedShow answer & explanation →
Q45.
The general solution of a non-homogeneous linear DE is:
A Just the particular integral, with little homogeneous contributionB Just the complementary function, mostly ignoring the forcing termC CF + PI (complementary function + particular integral)D The product of the complementary function and particular integralShow answer & explanation →
Q46.
The Bessel equation is:
A d<sup>2</sup>y/dx<sup>2</sup> + y = 0, the standard simple harmonic oscillator equationB x²d<sup>2</sup>y/dx<sup>2</sup> + xy_prime + (x²-n²)y = 0C d<sup>2</sup>y/dx<sup>2</sup> - y = 0, whose solutions are hyperbolic functionsD d<sup>2</sup>y/dx<sup>2</sup> + xy = 0, a form resembling the Airy equationShow answer & explanation →
Q47.
Euler-Cauchy equation x<sup>n</sup> y<sup>n</sup> + ... has solutions of the form:
A e<sup>mx</sup>, the solution form for constant-coefficient equationsB x<sup>m</sup> (power function)C sinx, a trigonometric solution formD xe<sup>x</sup>, a solution form for repeated rootsShow answer & explanation →
Q48.
d²y/dx² + 4y = cos 2x. The particular integral contains resonance because:
A The frequency on the right side happens to match some particular valueB cos 2x frequency matches the homogeneous solution frequency (2)C The coefficient of y in the equation happens to equal 4D The equation is second order, which often happens to involve resonanceShow answer & explanation →
Q49.
The existence and uniqueness theorem for y' = f(x,y) with y(x₀) = y₀ requires f to be:
A Constant in both x and y near the initial pointB Continuous and Lipschitz in y (near the initial point)C Periodic in x with some fixed, known periodD Polynomial in both x and y throughout the domainShow answer & explanation →
Q51.
The Green function approach solves:
A Non-linear DEs by linearizing the forcing term beforehandB Linear non-homogeneous DEs with arbitrary forcing using superpositionC Homogeneous DEs mainly, where the forcing term is already zeroD The Laplace equation specifically, rarely extended to other DE typesShow answer & explanation →
Q56.
The solution of the differential equation dy/dx = ky is:
A y = C·e<sup>kx</sup>B y = kx + CC y = C/x + kD y = k/x + CShow answer & explanation →
Q58.
An equation of the form dy/dx + Py = Q (P, Q functions of x) is called a ___ differential equation:
A linearB quadraticC homogeneousD exactShow answer & explanation →
Q59.
The integrating factor of the differential equation dy/dx + y = x is:
A e<sup>x</sup>B e<sup>−x</sup>C xD 1Show answer & explanation →
Q60.
The differential equation of the family of lines y = mx (m arbitrary) is:
A dy/dx = y/xB dy/dx = x/yC dy/dx = xD dy/dx = 1Show answer & explanation →
Q62.
The order and degree of (d²y/dx²)³ + (dy/dx)² + y = 0 are:
A order 2, degree 2B order 2, degree 3C order 3, degree 2D order 2, degree 1Show answer & explanation →
Q63.
The integrating factor of dy/dx + 2y = e<sup>x</sup> is:
A e<sup>x</sup>B e<sup>2x</sup>C e<sup>−2x</sup>D 2xShow answer & explanation →
Q64.
The general solution of dy/dx = e<sup>x − y</sup> is:
A e<sup>y</sup> = e<sup>x</sup> + CB e<sup>−y</sup> = e<sup>x</sup> + CC y = e<sup>x</sup> + CD e<sup>y</sup> = e<sup>−x</sup> + CShow answer & explanation →
Q65.
The general solution of dy/dx + y·tanx = secx is:
A y = cosx + C sinxB y = sinx + C cosxC y = tanx + CD y = secx + CShow answer & explanation →
Q66.
The differential equation whose general solution is y = A cosx + B sinx is:
A y'' − y = 0B y'' + y = 0C y' + y = 0D y'' + y' = 0Show answer & explanation →