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Differential equations questions in JEE are rarely hard once you know which method to use. The difficulty is the first ten seconds: looking at an equation and deciding whether it is separable, homogeneous or linear. This post gives you a short decision checklist, four worked examples, and a habit that catches most wrong answers.
Which types does JEE ask for?
The JEE (Main) 2026 syllabus lists order and degree, "separation of variables", homogeneous equations and linear equations of the type dy/dx + p(x)y = q(x). The JEE (Advanced) 2026 brochure adds the formation of ordinary differential equations and lists the same three solution methods. NCERT's Class 12 chapter on differential equations covers all three. Check the documents for your own exam year.
Why is solving not the same as understanding?
Arslan (2010) gave 77 university students 13 questions on differential equations: six on algebraic solution and the rest on interpreting equations and their solutions. Students who were "quite successful in algebraic solutions" did not fully understand the related concepts and had serious difficulties with them.
For you, that means two things. Practise recognising the type, not only carrying out the method. And understand what a solution is: a function that makes the equation true. That gives you a free check on every answer.
How do you spot the type?
Rewrite the equation as dy/dx = f(x, y), then ask in order:
Can I split it into g(y) dy = h(x) dx? Then it is separable. Integrate both sides.
Does f depend only on y/x? Test by replacing x with λx and y with λy. If f does not change, the equation is homogeneous. Put y = vx.
Is it of the form dy/dx + P(x)y = Q(x)? Then it is linear. Multiply by the integrating factor e^(∫P dx).
None of these? Try swapping roles. dx/dy + P(y)x = Q(y) is linear in x.
Worked example 1: separable
The number of undecayed nuclei satisfies dN/dt = −λN. Separate: dN/N = −λ dt. Integrate: ln N = −λt + C. With N = N₀ at t = 0, N = N₀e^(−λt). Setting N = N₀/2 gives the half-life, t = ln 2/λ. This is the same equation behind first-order kinetics and RC discharge in physics.
Worked example 2: homogeneous
Solve dy/dx = (x² + y²)/(2xy).
Replacing x and y by λx and λy leaves the right side unchanged, so put y = vx, which makes dy/dx = v + x dv/dx.
v + x dv/dx = (1 + v²)/(2v)
x dv/dx = (1 − v²)/(2v)
2v dv/(1 − v²) = dx/x
−ln|1 − v²| = ln|x| + c, so x(1 − v²) = K
Replace v = y/x: x² − y² = Kx.
Check: differentiate to get 2x − 2y(dy/dx) = K. Substitute K = (x² − y²)/x and solve for dy/dx. You get (x² + y²)/(2xy), as required.
Worked example 3: linear
Solve dy/dx + y/x = x² for x > 0.
Here P = 1/x, so the integrating factor is e^(ln x) = x. Multiply through: x(dy/dx) + y = x³, and the left side is d(xy)/dx. Integrate: xy = x⁴/4 + C, so y = x³/4 + C/x.
Check: dy/dx = 3x²/4 − C/x², and y/x = x²/4 + C/x². They add to x².
Worked example 4: linear in x
Solve dy/dx = y/(2x + y²).
It is not linear in y, but flip it: dx/dy = (2x + y²)/y = 2x/y + y, so dx/dy − (2/y)x = y. Now the integrating factor is e^(−2 ln y) = 1/y². Then d(x/y²)/dy = 1/y, so x/y² = ln|y| + C and x = y² ln|y| + Cy².
Which mistakes cost the most marks?
Reading P before the equation is in standard form. In x(dy/dx) + 2y = x³, divide by x first: P = 2/x, not 2.
Sign slips in the integrating factor. For dy/dx − 3y = e^x, P = −3 and the integrating factor is e^(−3x).
Losing the constant, or adding it only on one side after the integration is finished.
Giving a degree when it is not defined. Degree is defined only when the equation is a polynomial in its derivatives. An equation containing sin(dy/dx) has no degree.
Forcing a method. If the substitution test for homogeneity fails, the equation is not homogeneous, however much it looks it.
If the integration itself is slowing you down, our guide to the calculus JEE physics uses has worked examples of the integrals that come up most.
A practice routine
Sort before you solve (10 minutes, three times a week): take ten equations and write only the type and the first step for each. Do not solve them.
Mix the types. In Rohrer and Taylor's 2007 study, students who practised blocked problems scored 89% during practice but only 20% a week later; those who practised mixed problems scored 60% and then 63%. Mixed sets train the choice of method, which is the skill this chapter tests.
Check every answer by substitution, as in the examples.
Connect to physics. Once a week, solve one DE from kinetics, cooling or circuits, so the method meets a context. Our post on why theory does not always become numericals explains why that link matters.
When you are stuck on the first step, Fermi's AI tutor can work out what you already know and build a plan for the session, so you work on recognising the type rather than only copying a method.
Tonight, sort ten differential equations from your textbook exercise into separable, homogeneous and linear before solving any of them.
