Subject workflows · Updated 13 September 2026

How to Study Differential Equations With AI (Safely)

Solving the equation is the one part of this course you never have to take on trust. That is exactly why you should stop spending your study time on it.

A differential equations course asks you to do three separate things: turn a situation into an equation, solve the equation, and say what the solution means. A chatbot does the middle one. The middle one also happens to be the only one that can be checked mechanically in about ten seconds — you substitute the answer back in and see whether it reduces to zero. So the sensible split is the opposite of how most people use these tools: let software do the solving and verify it by substitution rather than by reading it, and spend your own effort on the modelling and the interpretation, which are unverifiable, harder, and where the marks are.

The course is three jobs, not one

MIT states it directly in the description of 18.03SC Differential Equations: "The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions."

Model, solve, interpret. Almost every hour students spend on this course goes into the second verb, because it is the one that looks like work — integrating factors, characteristic equations, undetermined coefficients, partial fractions for a Laplace transform. But the syllabus is built around all three. MIT's four units run from first-order equations and their geometric and numerical methods, through second-order constant-coefficient equations and damped oscillators, into Fourier series and Laplace transforms, and end at first-order systems and phase portraits — and the whole of that last unit is interpretation with almost no closed-form solving in it. The full course, including problem sets and exams, is free under a Creative Commons licence, as are Paul Dawkins' Differential Equations notes, which follow the same spine.

Solving is the part you never have to trust

Here is what makes this course unusual. In most maths, checking an answer costs about as much as producing it. In differential equations it is nearly free, because a solution is a claim that can be tested: put the function back into the equation, differentiate, simplify, and see whether both sides agree. If they do, it is a solution — regardless of who or what produced it, and regardless of how confident the working sounded.

This is built into the tools. SymPy, the free Python computer algebra library, ships a function called checkodesol whose documentation describes it in one line: it "substitutes sol into ode and checks that the result is 0", returning a tuple whose first item is True if the substitution results in zero "and False otherwise". It will tell you the same thing about an answer a chatbot wrote, an answer you wrote at 1am, and an answer from the back of the book.

Two honest caveats, because this is a proof of something slightly narrower than students assume. First, substitution shows you have a solution, not the general solution. An nth-order equation needs n independent arbitrary constants; a model that quietly loses one produces an expression that passes substitution and is still incomplete, which is a common way to drop marks. Count the constants against the order every time. Second, the initial or boundary conditions are a separate check — substitute those too, into the solution itself rather than into the equation.

Where marks are actually lost: naming the equation

The reputation is old. Rezvanifard, Radmehr and Rogovchenko, writing in Teaching Mathematics and its Applications in 2023, open their study of engineering students and exact equations by quoting Sazhin's 1998 verdict that "differential equations, even relatively simple ones, seem to be a stumbling block for many students."

The interesting question is which step is the stumbling block, and the same paper reports a result from Arslan that answers it. Given an equation that was both exact and homogeneous — solvable by either of two recognisable routes — 96% of students solved it correctly. Given a purely exact equation, correct solutions fell to 77%, because, in the paper's words, "several students could not identify the type of equation and solution method."

Nothing about the algebra got harder between those two problems. What changed is that the second one required the student to identify a single type before starting. That is the bottleneck: not integrating, but classifying. Is it first order or second? Linear or nonlinear? Separable, exact, homogeneous, Bernoulli? Autonomous or not? Constant coefficients or variable?

Which is precisely the step a chatbot deletes when you paste an equation and ask it to solve. It classifies silently, in a fraction of a second, and hands you the worked method — so the one decision your exam is built around is the one you never practise. Notice that SymPy makes the same decision explicit rather than hidden: classify_ode returns "a tuple of possible classifications for an ODE", ordered so the first is what dsolve would try by default. Use that to check a classification you already committed to, and it is a drill. Use it first, and you have skipped the exam.

The solution is a function, not a number

The third job, interpretation, has its own documented failure mode. The same 2023 paper summarises Chris Rasmussen's foundational 2001 study of students in a differential equations course as documenting "difficulties and confusion arising from the interpretation of solutions as functions rather than numbers."

That habit is imported from every previous maths course, where solving meant finding a value of x. Here the answer is a whole family of curves, and the questions that follow are about shape: what happens as t grows, whether the system settles or blows up, where the equilibrium solutions sit and whether a nudge away from one comes back or runs away. Equilibria and stability are features of autonomous equations specifically — a detail worth keeping straight, because the test is not simply "where is the derivative zero".

A text chatbot answers these in prose, which is the format least likely to fix the misconception. A picture works better, and it is free: GeoGebra's SlopeField command "plots a slope field of the differential equation dy/dx=f(x,y)", and the companion SolveODE command traces a particular solution numerically from a start point you hand it. Type your equation, define a free point, pass it to SolveODE, then drag it around the field. Watching the curve change as the initial condition moves is the clearest demonstration there is that the solution is a function and the constant is a choice.

Split the job

The workflow, step by step

Four prompts. Paste them into ChatGPT, Claude, Gemini or whatever you already use.

1. Make it test your classification instead of solving. This is the step your exam is built on, so protect it.

You are my differential equations classification coach. Here is an
equation from my problem set:

[paste the equation]

Do NOT solve it and do NOT name the method.

Ask me one question at a time, waiting for my answer each time:
1. What is the order, and is it linear or nonlinear?
2. Is the independent variable absent - is it autonomous?
3. Which standard form does it match: separable, linear first order,
   exact, Bernoulli, homogeneous, constant-coefficient, or none?
4. What test did I apply to decide that, and what was the result?

If I am wrong, say only which test I applied incorrectly. Do not give
me the right classification. When I have it, confirm and stop - I
will solve it myself.

2. Build the model out loud. Word problems — mixing tanks, cooling bodies, RC circuits, populations — are where the modelling mark lives.

Here is a word problem I have to turn into a differential equation:

[paste the problem]

Do NOT write the differential equation and do NOT solve it.

Ask me one question at a time until I have stated:
- what quantity is changing, and what it is a function of
- the units of that quantity and of its rate of change
- every process adding to it and every process removing from it
- the initial condition, and where the problem states it

Then check exactly two things about the equation I write: whether the
units match on both sides, and whether every term corresponds to a
process I named. Report mismatches without fixing them.

3. Verify by substitution, in code. Do not accept an answer that was reasoned out in prose — including your own.

Verify this solution by substitution using code, not by reasoning it
out. Use SymPy.

1. Define the ODE and my candidate solution as SymPy expressions.
2. Run checkodesol() on them and show me the raw output tuple.
3. Separately substitute my initial condition into the solution and
   show the result.
4. Tell me how many arbitrary constants my solution contains and
   whether that matches the order of the equation.

Show the code you ran and its unedited output. If you cannot execute
code in this chat, say so and stop. Do not estimate, and do not
simplify anything by hand between the code and your answer.

ODE: [paste]
My solution: [paste]
Initial condition: [paste]

ChatGPT, Gemini and Claude all have some form of code execution, but what is available shifts with plans and interfaces — which is why the prompt tells it to stop rather than improvise. If it stops, or you cannot tell whether it truly ran anything, do the substitution by hand; on a first-order equation that is a two-line differentiation. The habit generalises, and our guide to verifying AI answers before you study from them applies it elsewhere.

4. Interpret it. You have a function. Now say what it does.

I have solved this equation and I want to check that I understand
what the answer means, not whether the algebra is right.

ODE: [paste]
My solution: [paste]

Ask me, one at a time:
1. What does my solution do as t gets large, and why?
2. Which constants came from the physical set-up and which came from
   the initial condition?
3. If the equation is autonomous, where are the equilibrium solutions
   and is each one stable or unstable?
4. If I doubled [name one parameter], what changes in the graph?

Do not answer any of these for me. Tell me only whether my answer is
right, and if it is wrong, name the fact I have got backwards.

Where the line is

The integrity rule here is unusually clean, because the three jobs separate so neatly. Having a model produce the equation, the method and the answer for a graded problem set substitutes its work for yours under almost any policy, and pasting an active assignment into a public tool can breach the policy before anyone marks it. Using a CAS to check a solution you derived, having a chatbot quiz you on classification, or asking it to explain a method your lecture rushed is ordinary studying — and on many courses a CAS is expected. Read your syllabus, and if the wording is vague, our guide to homework help without cheating and the class AI policy checklist cover how to read it and how to ask in writing.

Related reading

• The courses either side. College calculus supplies the integration this course leans on, and multivariable calculus runs alongside it.
• The method that gets its own chapter. Laplace transforms and system stability is how the second half of most courses handles forcing and discontinuous inputs.
• Where systems of equations come from. Phase portraits are eigenvalue problems — see eigenvalues and eigenvectors.
• Where it is applied. Control systems and transfer functions and circuit analysis are this maths with units attached.
• For the midterm. The STEM problem-set workflow turns this into a revision schedule.

FAQ

Can ChatGPT solve differential equations correctly?

Often, and it does not matter very much, because you can check it outright instead of guessing. Substitute the answer back into the equation: if both sides agree, it is a solution no matter what produced it, and if they do not, it is wrong no matter how confident the working looked. SymPy's checkodesol function does exactly this substitution for free. The one thing substitution will not catch is a missing arbitrary constant, so also count the constants against the order of the equation, and check any initial condition separately.

What is the hardest part of a differential equations course?

Recognising which type of equation you are looking at. Research by Arslan, reported in a 2023 study by Rezvanifard, Radmehr and Rogovchenko, found that 96% of students correctly solved an equation that was both exact and homogeneous, while only 77% solved a purely exact equation, because several students could not identify the type of equation and solution method. The algebra was no harder; the second problem simply required them to classify it first. That is the step a chatbot removes when you paste an equation and ask it to solve.

What is the best free tool for drawing a slope field?

GeoGebra, which runs in a browser at no cost. Its SlopeField command plots a slope field of the differential equation dy/dx=f(x,y), and the SolveODE command draws a particular solution curve through a point you place. Dragging that point around the field shows the whole family of solutions changing as the initial condition changes, which is the clearest way to internalise that the answer is a function rather than a number.

Why do I keep losing marks even though my answer checks out?

Usually because the answer is a solution but not the general solution. An nth-order equation needs n independent arbitrary constants, and an expression that has lost one will still pass a substitution check while being incomplete. The other common cause is the interpretation question that follows: long-run behaviour, equilibrium solutions and their stability, or what a parameter physically means. Those follow-up questions are built to find exactly the gap Chris Rasmussen documented in 2001, when students treated solutions as numbers rather than functions.

Is it cheating to use AI for differential equations homework?

It depends on your course policy, but the three jobs separate cleanly. Having a model write the equation, choose the method and produce the answer for work you submit is substituting its effort for yours under almost any policy. Checking a solution you derived, drilling classification, or asking for an explanation of a method is ordinary studying, and many courses expect you to use a computer algebra system anyway. Read the syllabus, and ask your instructor in writing when it is unclear.

Bottom line

Differential equations is the rare course where the answer proves itself, so treat the solving step as settled admin: hand it to software and verify it by substitution rather than by reading the working. Spend the time you get back on the two jobs nothing can check for you — writing the equation from the situation, and saying what its solution does. Classify before you solve, count your constants against the order, and look at the slope field until the family of curves feels obvious.