NDA Maths · Teaching notes
Differential Equations — NDA Mathematics
Differential Equations is a steady 63-PYQ chapter (2017–2026), ~29% HARD, built on classification plus a fixed toolkit of solving methods. The chapter teaches in three movements, ordered so each builds on the last: (1) Order, degree, and solutions — how to classify an ODE (order = highest derivative, degree = power of that derivative after clearing radicals) and what a 'solution' means (the number of arbitrary constants equals the order); (2) Formation — given a family of curves with arbitrary constants, differentiate to eliminate the constants and recover the ODE; (3) Solving and verifying — the methods that actually integrate an ODE: separating variables, reducing by substitution, the integrating factor for linear equations, and the growth/decay and initial-value applications. 8 concepts, every PYQ tagged. Many questions only ask for order/degree — fast marks — while the HARD ones reward knowing which solving method the equation's shape calls for.
Subtopic notes
Order, Degree and Solutions
22 PYQsThe order of a differential equation is the highest derivative present; the degree is the power of that highest derivative once the equation is made polynomial in its derivatives; the number of arbitrary constants in a solution equals the order.
Open note
Forming an ODE from a Family of Curves
12 PYQsTo form the differential equation of a family of curves, differentiate enough times to eliminate every arbitrary constant — n constants need n differentiations and produce an order-n equation.
Open note
Solving ODEs — Separable, Substitution, Integrating Factor
29 PYQsFirst-order ODEs are solved by a small toolkit: separate the variables, reduce a tangled one by substitution, or use an integrating factor for the linear case — then fit any initial condition.
Open note
PYQ weightage by concept
8 concepts · 63 PYQs — where the marks actually sit, so you know what to drill first
PYQ weightage by concept
8 concepts · 63 PYQs — where the marks actually sit, so you know what to drill first
| Concept | PYQs | Share |
|---|---|---|
| Order and degree of a differential equation | 18 | 29% |
| Solutions and arbitrary constants | 4 | 6% |
| Concept | PYQs | Share |
|---|---|---|
| Eliminating arbitrary constants | 9 | 14% |
| Matching an ODE to its general solution | 3 | 5% |
| Concept | PYQs | Share |
|---|---|---|
| Separation of variables | 10 | 16% |
| Initial-value problems and growth/decay | 9 | 14% |
| Reducible to separable by substitution | 7 | 11% |
| Linear equations and the integrating factor | 3 | 5% |
Formula & revision sheet
1 formulas · 8 gotchas across all subtopics — the exam-eve cheat-sheet
Formula & revision sheet
1 formulas · 8 gotchas across all subtopics — the exam-eve cheat-sheet
Watch out for (2)
- Clear fractional powers BEFORE reading the degree→ Order and degree of a differential equation
- Count INDEPENDENT constants→ Solutions and arbitrary constants
Watch out for (2)
- Differentiate as many times as there are constants→ Eliminating arbitrary constants
- A circle needs equal squared-term coefficients→ Matching an ODE to its general solution
Watch out for (4)
- Take logs / exponentials to unlock separation→ Separation of variables
- Spot the glued combination first→ Reducible to separable by substitution
- If it is not linear in y, try linear in x→ Linear equations and the integrating factor
- Apply the initial condition to the GENERAL solution→ Initial-value problems and growth/decay