What the board asks

We went through the ISC board papers, question by question, and matched each question to a topic. Here is what was asked, how often, and what was never asked.

Mathematics: 309 topics in the 2027 syllabus, across 11 units. The board has asked 255 of them at least once. 54 were never asked in the papers we checked.

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M7 Calculus: Differential Equations

All the topics in Mathematics: Calculus: Differential Equations

Asked topics first, the most asked at the top. Tap a topic to see what to write, the result, the years, and the real board questions.

18 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Linear DE dy/dx + Py = Q: general solutionReduce to standard form, write P and Q, find the IF, write y (IF) = integral of Q (IF) dx + C, integrate, and simplify.	Sum	5	3 papers	2026	Asked
2	Linear DE in x: dx/dy + Px = QRecognise that the equation is linear in x, rewrite as dx/dy + Px = Q with P, Q functions of y, find IF = e^(integral of P dy), write x (IF) = integral of Q (IF) dy + C, and use a condition if given.	Sum	5	3 papers	2026	Asked
3	Variables separable: general solutionFactorise the right side if needed, separate the variables with all y on one side and all x on the other, integrate both sides, and add one constant C.	Sum	3	3 papers	2025	Asked
4	One-line analysis of a separable equation's solutionSeparate and integrate quickly, then compare the solution with the given condition (a circle, a curve through a point, an MCQ option) and state the value or option with a one-line reason.	Give reason	1	2 papers	Specimen 2027	Asked
5	Homogeneous DE in x/y: substitute x = vyWhen the equation is a function of x/y (for example contains e^(x/y)), write it as dx/dy, put x = vy and dx/dy = v + y dv/dy, separate, integrate, and replace v by x/y; use a given condition if there is one.	Sum	5	2 papers	Specimen 2027	Asked
6	Variables separable: particular solution from a conditionSeparate, integrate (using partial fractions or a standard integral where needed), add C, substitute the given values to find C, and write the particular solution.	Sum	3	2 papers	2026	Asked
7	Homogeneous DE: substitute y = vx, general solutionShow the equation is homogeneous, put y = vx and dy/dx = v + x dv/dx, separate v and x, integrate, and replace v by y/x.	Sum	5	2 papers	2026	Asked
8	Homogeneous DE: particular solution from a conditionSolve by y = vx as in the general case, then substitute the given point to find the constant and write the particular solution.	Sum	5	2 papers	2025 improvement	Asked
9	Form the DE of a family with one arbitrary constantDifferentiate the family once, eliminate the constant using the original equation, and write the differential equation without the constant.	Sum	2	2 papers	2020	Asked
10	Classify a first-order DE and choose the methodTest in order whether the equation is separable, reducible by substitution, homogeneous, or linear in y or in x, and state the method with a one-line reason.	Give reason	1	1 paper	2026	Asked
11	Verify that a given function is a solutionDifferentiate the given function (or solve the equation), substitute into the differential equation, and show both sides agree; for a general solution, note the number of arbitrary constants.	Proof	2	1 paper	2025	Asked
12	Test whether a differential equation is homogeneousWrite dy/dx = F(x, y), replace x by tx and y by ty, and check whether F(tx, ty) = F(x, y); state the conclusion for each option.	Give reason	1	1 paper	2025	Asked
13	Find the integrating factor of a linear equationDivide to bring the equation to dy/dx + Py = Q, identify P, and write IF = e^(integral of P dx), simplified.	Statement	1	1 paper	2025 improvement	Asked
14	Linear DE in y: particular solution from a conditionSolve the linear equation in full, then substitute the given values to find C and write the particular solution.	Sum	5	1 paper	2025 improvement	Asked
15	Order and degree of a differential equationWrite the highest derivative present (order), clear radicals and fractions in the derivatives, and state the power of the highest derivative (degree), or say the degree is not defined if the equation is not a polynomial in derivatives.	Statement	1	1 paper	2024	Asked
16	Reducible to separable: substitution v = ax+by+cPut v equal to the repeated linear expression, write dv/dx in terms of dy/dx, rewrite the equation in v and x, separate, integrate, and substitute back.	Sum	3	1 paper	2019	Asked
17	Word problem: growth or decay by a differential equationTranslate the rate statement into dP/dt = kP, separate and integrate to P = P0 e^(kt), use the given data to find k, and answer the time or amount asked with units.	Sum	3	1 paper	2018	Asked
18	Form the DE of a family with two arbitrary constantsDifferentiate twice, use the three equations to eliminate both constants, and write the resulting second-order equation.	Sum	2	never		Not in syllabus

This list was made from the 2027 syllabus and checked with 10 board papers, including the 2027 specimen paper. It is not yet checked with the textbook.

The 10 topics Mathematics papers ask most

Counted on the board papers from 2018 to the 2027 specimen paper, which every unit has been checked against.

1.Formulate an LPP from a word problemM10 Linear Programming · asked 7 times since 2018
2.Scalar and vector projection of one vector on anotherM8 Vector Algebra · asked 6 times since 2018
3.Mean (expected value) of a random variableM11 Probability · asked 6 times since 2018
4.Find the inverse of an invertible algebraic function and verifyM1 Relations and Functions · asked 5 times since 2018
5.Principal value of expressions and nested inverse trigM1 Relations and Functions · asked 5 times since 2018
6.Evaluate or show zero without expanding, using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
7.Prove a determinant equals a factorised expression using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
8.Definite integral of a modulus functionM5 Calculus: Integrals · asked 5 times since 2018
9.Case study or linked parts on lines and planesM9 Three-Dimensional Geometry · asked 5 times since 2018
10.Equations solved with the complementary identitiesM1 Relations and Functions · asked 4 times since 2018

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