Differential Equations Course for CSIR NET and GATE Maths

You need a course whose weekly plan matches the exam's list: existence and uniqueness, variable-coefficient equations, Sturm-Liouville problems, Green's function and PDE methods. A first course built for engineers teaches techniques well. By our count it meets only 2 of the 12 items on CSIR NET's list.

The decision that matters is not free versus paid but whether the contents match. Take any course's week-by-week plan, lay it beside the table in the next section, and tick what it teaches. Whatever stays unticked, you will have to learn from a book or a teacher.

CSIR NET's common syllabus for Parts B and C lists 7 ODE items and 5 PDE items in Unit 3, beside numerical analysis, calculus of variations, integral equations and classical mechanics. GATE's list is longer: 15 ODE items in Section 5 and 10 PDE items in Section 9. It also names Euler and Runge-Kutta methods under Numerical Analysis. Parts B and C are the subject sections of the paper, set out in our CSIR NET exam pattern guide.

We marked an item as taught in a first course when the MIT 18.03 topic list names it. That list covers:

• first-order equations
• linear equations, especially second order with constant coefficients
• undetermined coefficients and variation of parameters
• Fourier series
• Laplace transforms
• matrix systems
• nonlinear autonomous systems

Linear theory counts as covered for constant coefficients only. These are our counts for comparison, not official figures, and splitting the phrases differently moves the totals slightly.

Terms used in the table and the lists after it:

• Initial value problem: an ODE plus the value of the unknown function at one starting point.
• Boundary value problem: an ODE plus conditions at two points, such as y(0) = y(1) = 0.
• Existence and uniqueness: whether a solution exists and whether it is the only one.
• Singular solution: a solution that no choice of constants in the general solution produces.
• Sturm-Liouville problem: a boundary value problem with a parameter λ. Its special values (eigenvalues) give non-zero solutions (eigenfunctions).
• Green's function: the function that gives the solution of a boundary value problem for any right-hand side.
• Lagrange-Charpit method and method of characteristics: ways to solve first-order PDEs by following curves along which the PDE becomes an ODE.
• Cauchy problem: a PDE with data given along a curve.
• Lipschitz condition: the right-hand side f(x, y) changes in y at most a fixed multiple of the change in y.
• Cauchy-Euler equation: an equation of the form x²y″ + axy′ + by = 0.
• Frobenius method: power-series solutions near a regular singular point.
• Lyapunov function: an energy-like function used to prove stability.
• Legendre and Bessel equations: named ODEs whose solutions are special functions.

Two of the 12 CSIR items are first-course material (rows 3 and 5), so ten are gaps.

On GATE's 15 ODE items, seven are first-course material:

• first-order equations
• higher-order equations with constant coefficients
• the definition and properties of Laplace transforms
• Laplace transforms applied to ODEs
• systems of linear first-order equations
• stability of stationary points for linear systems
• linearised stability

The other eight are gaps:

• existence and uniqueness
• variable-coefficient equations
• the Cauchy-Euler equation
• series solutions (power series and the Frobenius method)
• Legendre and Bessel functions
• Sturm's oscillation and separation theorems (statements about where solutions cross zero)
• Sturm-Liouville problems
• Lyapunov functions

All 10 PDE items are gaps:

• method of characteristics
• classification and canonical forms
• separation of variables
• maximum principle
• heat and wave equations in one space variable
• the wave Cauchy problem and d'Alembert formula (the explicit solution of the one-dimensional wave equation)
• domains of dependence and influence
• the non-homogeneous wave equation
• the heat equation's Cauchy problem
• Laplace and Fourier transform methods

That makes 7 covered and 18 gaps out of 25.

Six items sit on both lists: existence and uniqueness, systems of first-order equations, the general theory of linear equations, Sturm-Liouville problems, classification of second-order PDEs and separation of variables. Two more, first-order PDE methods and the Cauchy problem, are the same family under different names (Lagrange-Charpit on CSIR NET, characteristics on GATE).

Preparing for one exam therefore covers most of the shared core for the other. The remainder differs:

• GATE adds Laplace transforms, series solutions, Legendre and Bessel functions, stability theory, the maximum principle, and the wave and heat equation theory.
• CSIR NET adds singular solutions, Green's function, variation of parameters and higher-order constant-coefficient PDEs.

The Indian Institute of Technology Joint Admission Test for Masters (IIT JAM) is a separate exam with its own syllabus, set out beside GATE's in our GATE and IIT JAM Mathematics syllabus comparison.

A first course is written for students who will use equations, not prove things about them. It trains you to recognise a type and apply a recipe. The exam items above ask when a solution exists, whether it is the only one, and how eigenvalues and boundary conditions pin it down. Those are theory questions, and a recipe cannot answer them.

This also explains a common frustration: you can follow every worked example but freeze on a fresh problem. Chapters teach methods one at a time, so you always know which method is coming. An exam never names the chapter. In our view, the fix is to learn classification first, meaning how to name the type of equation before you touch it. Then practise by type until the method follows automatically.

Textbooks split the same way. A gentle text is the right start for techniques, and a rigorous one is needed for existence, uniqueness and Sturm-Liouville theory. Beginners who open the rigorous one first tend to stall, so work in that order. Our guide to CSIR NET maths books covers which books to use.

Run these six closed-book tests, one for each of six of the ten CSIR NET gaps. Each has a pass mark, and every failed test names a topic to study before anything else. Attempt each with no notes.

Our recommendation for planning: count the ODE and PDE questions in the last five CSIR NET or GATE papers, for example from our guide to GATE Mathematics previous papers with solutions. If they make up more than a fifth of the marks, give them at least that share of your study time.

Free courses are enough for solution techniques, and a paid course earns its fee on the gaps. MIT 18.03 is free. So are the National Programme on Technology Enhanced Learning (NPTEL) courses Ordinary Differential Equations and Ordinary and Partial Differential Equations and Applications. Those links are to the 2024 runs, so look for the current run and check its weekly plan against the table.

We also publish free mathematics study content and a PDF Drive with notes and previous-year questions.

The free route works if you can study the theory from a book and mark your own work. A paid course is worth considering when you get stuck and need an answer on Green's function the same week. It is also worth it when you want tests that mirror the paper. We publish this page and we sell a course, so check our view against the table, which you can verify yourself.

Our CSIR-NET and GATE Mathematics 2027 course teaches Ordinary Differential Equations and Partial Differential Equations as separate subjects. Our free differential equations series on our Telegram channel shows how we teach them. It has lectures on constant coefficients, Cauchy-Euler equations, variable coefficients, systems of equations, regular singular points, Legendre and Bessel equations, Sturm-Liouville problems, Green's function, and the Cauchy problem for linear and non-linear PDEs. Ask us to confirm coverage of existence and uniqueness, singular solutions, PDE classification and separation of variables before you enrol.

As of October 2026 the price is ₹15,999 one time, or four instalments of ₹3,999.75 (₹15,999 in total), with access through 30 June 2027. The course is live online with recorded lectures for revision, in Hindi and English. It includes:

• handwritten notes and PDFs
• assignments
• previous-year question (PYQ) discussion
• mock tests
• weekly doubt sessions or personal mentoring, with continuous doubt support

SB Sir (Sunil Bansal) designs the test series and conceptual tests. He secured All India Rank 11 in CSIR NET in December 2017 and is GATE MA qualified.

The batch began on 10 July 2026, and its recorded lectures are available for revision. If you sit CSIR NET in December 2026, our CSIR-NET December 2026 Live Batch is ₹11,999 one time, or three instalments of ₹3,999.67, with access till January 2027. It lists Differential Equations among its nine subjects. To plan how to catch up on earlier classes, contact us between 9am and 7pm, Monday to Saturday.

To judge the teaching before you commit, start with the ₹999 demo plan on the 2027 course page.