Last Updated: May 2026
JEE Main Differential Equations 2027 (Class 12 Maths Chapter 9) yields 2–3 questions per paper, weighted at 8–12 marks. This chapter combines integration, algebra and physics-style modelling in a single equation. Top JEE Main scorers solve ALL three difficulty tiers — variable separation, linear, and homogeneous — under 90 seconds each. This 2,000-word JEE Main Differential Equations 2027 guide covers definitions, all standard methods, and 40 practice problems with worked solutions for the harder ones.
1. Definitions — Order and Degree
Order = highest derivative order in the equation. For (d²y/dx²)³ + (dy/dx) = sin x, order = 2.
Degree = highest power of the highest-order derivative AFTER the equation is made polynomial in derivatives (free of fractions, radicals). In above example, degree = 3.
Note: Degree is undefined when the equation cannot be made polynomial (e.g., cos(dy/dx) = x).
2. Solution: General vs Particular
General solution contains arbitrary constants equal to the order. e.g., y = c1·cos x + c2·sin x is the general solution of d²y/dx² + y = 0.
Particular solution is obtained by giving specific values to the constants using initial/boundary conditions.
3. Method 1 — Variable Separable
If dy/dx = f(x)·g(y), separate: dy/g(y) = f(x)dx, then integrate both sides.
Example: dy/dx = (1+y²)/(1+x²) → dy/(1+y²) = dx/(1+x²) → tan⁻¹y = tan⁻¹x + C.
4. Method 2 — Homogeneous Equations
If dy/dx = f(y/x), substitute y = vx, dy/dx = v + x(dv/dx). Reduces to variable separable in v and x.
Example: dy/dx = (x+y)/(x-y). Put y=vx → v + x(dv/dx) = (1+v)/(1-v). Solve for v in x; replace v=y/x.
5. Method 3 — Linear (First Order)
Standard form: dy/dx + Py = Q (P, Q functions of x).
Integrating factor (IF) = e^∫P dx. Solution: y·IF = ∫(Q·IF)dx + C.
Example: dy/dx + y = e^x. P=1, IF=e^x. y·e^x = ∫e^(2x)dx = e^(2x)/2 + C → y = e^x/2 + C·e^(-x).
6. Method 4 — Bernoulli’s Equation
Form: dy/dx + Py = Q·y^n. Divide by y^n, substitute t = y^(1-n) → reduces to linear.
7. Method 5 — Exact Differential Equations
M(x,y)dx + N(x,y)dy = 0 is exact iff ∂M/∂y = ∂N/∂x. Solution: ∫M dx (treating y constant) + ∫(terms of N free of x)dy = C.
8. Method 6 — Second-Order Linear with Constant Coefficients
Form: a·d²y/dx² + b·dy/dx + c·y = 0. Auxiliary equation: am² + bm + c = 0.
Cases: (i) Real distinct roots m1, m2: y = A·e^(m1x) + B·e^(m2x). (ii) Real repeated root m: y = (A + Bx)·e^(mx). (iii) Complex roots α±iβ: y = e^(αx)·(A·cos βx + B·sin βx).
9. Forming Differential Equations from Curves
Eliminate arbitrary constants by differentiating equation as many times as the number of constants, then substituting.
Example: Family of circles x² + y² = r² → 2x + 2y(dy/dx) = 0 → x + y(dy/dx) = 0.
10. Applications
| Application | Equation |
|---|---|
| Population growth | dN/dt = kN → N = N₀·e^(kt) |
| Radioactive decay | dN/dt = -λN → N = N₀·e^(-λt) |
| Newton’s cooling | dT/dt = -k(T-T₀) |
| Kirchhoff (RC circuit) | R(dq/dt) + q/C = V |
| SHM | d²x/dt² + ω²x = 0 |
11. 40 Practice Problems (Sample 10)
- Order and degree of (d²y/dx²)² + (dy/dx)³ + y = 0: (a) 2,2 (b) 2,3 (c) 3,2 (d) 1,3 — (a)
- Solution of dy/dx = e^(x+y): (a) e^(-y) + e^x = C (b) e^y – e^x = C (c) e^(-y) – e^x = C (d) e^y + e^x = C — (c)
- IF of dy/dx + y/x = x²: (a) x (b) x² (c) 1/x (d) e^x — (a)
- Solution of x(dy/dx) + y = x³: y·x = ? (a) x⁴/4 + C (b) x³/3 + C (c) x²/2 + C (d) x⁴/4 — (a)
- The substitution to make dy/dx = (x+y)² linear: (a) y = vx (b) x = vy (c) v = x+y (d) None — (c)
- Order of differential equation of family y = A·sin(x+B): (a) 1 (b) 2 (c) 3 (d) 0 — (b)
- Solution of d²y/dx² – y = 0: y = ? (a) A·cos x + B·sin x (b) A·e^x + B·e^(-x) (c) (A+Bx)e^x (d) e^x·(A·cos x + B·sin x) — (b)
- Number of arbitrary constants in general solution of n-th order ODE: (a) 1 (b) n (c) n+1 (d) 2n — (b)
- Equation x²·dy + y²·dx = 0 is: (a) Linear (b) Variable separable (c) Bernoulli (d) Exact — (b)
- Newton’s cooling involves: (a) Linear ODE (b) Bernoulli (c) Variable separable (d) Both (a) and (c) — (d)
12. Common Mistakes
- Computing degree before making the equation polynomial in derivatives.
- Forgetting + C in indefinite integration.
- Misusing IF formula on non-linear equations.
- Treating M dx + N dy = 0 as exact without checking ∂M/∂y = ∂N/∂x.
Frequently Asked Questions
Q1. JEE Main vs Advanced — difference in this chapter?
Main tests linear, variable separable, homogeneous (1–2 Qs). Advanced adds exact + 2nd-order constant-coefficient + applications (1–2 Qs).
Q2. NCERT enough for this chapter?
For Main — yes. For Advanced — supplement with Cengage or RD Sharma.
Internal Resources
- JEE Main Limits and Continuity
- JEE Main Coordinate Geometry
- JEE Main Permutations Combinations
- Free JEE Mock Test
[cg_quiz id=”jee-differential-eqs-2027″]