JEE Main Mathematics · Chapter 11

Differential Equations

Order and degree, variable separable, homogeneous and linear differential equations.

NCERT Class 12JEE Main: 1–2 Qs per shift 8 min read

1. Chapter at a Glance

A differential equation (DE) is an equation that involves an unknown function and its derivatives, such as dy/dx = ky. DEs describe growth, decay, cooling, motion and many other changing quantities. JEE Main asks one or two questions every shift, most often a linear DE or a homogeneous DE with an initial condition, followed by finding y at some point.

What JEE asks most
  • Solving first-order linear DEs using the integrating factor.
  • Variable separable and homogeneous DEs with an initial condition.
  • Order and degree of a given DE.
  • Forming the DE of a family of curves.

2. Order, Degree and Solutions

  • Order: the order of the highest derivative present.
  • Degree: the power of the highest-order derivative, after the equation is made free of radicals and fractions in the derivatives. The degree is not defined if the DE is not a polynomial in its derivatives (for example if it contains sin(dy/dx) or ey′).
  • A general solution contains as many arbitrary constants as the order. A particular solution is obtained by fixing the constants using given conditions.

Forming a differential equation

To form the DE of a family with n arbitrary constants, differentiate n times and eliminate the constants. So the order of the DE equals the number of independent arbitrary constants. For example y = Aeˣ + Be−x gives y″ = y (order 2).

3. Variables Separable

If the DE can be written as f(y) dy = g(x) dx, integrate both sides:

∫f(y) dy = ∫g(x) dx + C

Example: dy/dx = ex−y ⇒ ey dy = eˣ dx ⇒ ey = eˣ + C.

Reducible form: dy/dx = f(ax + by + c) — put t = ax + by + c.

Growth and decay

dP/dt = kP gives P = P0ekt (growth for k > 0, decay for k < 0). If a quantity doubles in time T, then k = (ln 2)/T and the time to become m times is T·(ln m)/(ln 2).

4. Homogeneous Differential Equations

dy/dx = f(x, y)/g(x, y) is homogeneous if f and g are homogeneous functions of the same degree — i.e. the right side can be written as a function of y/x alone.

Put y = vx ⇒ dy/dx = v + x dv/dx, then separate v and x

Example: dy/dx = (x + y)/x = 1 + v ⇒ v + x dv/dx = 1 + v ⇒ dv = dx/x ⇒ v = ln|x| + C ⇒ y = x ln|x| + Cx.

5. Linear Differential Equations

A first-order linear DE has the form

dy/dx + P(x) y = Q(x)

Multiply by the integrating factor IF = e∫P dx. The left side becomes the derivative of y·IF:

y · IF = ∫Q · IF dx + C

Similarly, dx/dy + P(y) x = Q(y) has IF = e∫P dy and solution x · IF = ∫Q · IF dy + C. Use this form when the DE is linear in x rather than in y.

Bernoulli form

dy/dx + Py = Qyⁿ is solved by dividing by yⁿ and putting t = y1−n, which makes it linear in t.

C = 1C = 0.5C = −1 O
The general solution y = Ceˣ of dy/dx = y is a whole family of curves; an initial condition picks out one of them.

6. Formula Master Table

TypeFormMethod
Variable separablef(y)dy = g(x)dxIntegrate both sides
Reducible to separabley′ = f(ax + by + c)t = ax + by + c
Homogeneousy′ = F(y/x)y = vx
Linear in yy′ + Py = QIF = e∫P dx
Linear in xx′ + Px = QIF = e∫P dy
Bernoulliy′ + Py = Qyⁿt = y1−n
Exact differentialsx dy + y dx = d(xy)Recognise and integrate
Growth / decaydP/dt = kPP = P0ekt

Useful exact differentials

  • x dy + y dx = d(xy)
  • (x dy − y dx)/x² = d(y/x); (y dx − x dy)/y² = d(x/y)
  • (x dy − y dx)/(x² + y²) = d(tan−1(y/x))
  • (x dx + y dy)/(x² + y²) = (1/2) d(ln(x² + y²))

7. Mind Map

Differential Equations
Basics
  • Order, degree
  • General vs particular
  • Forming DE of family
Separable
  • f(y)dy = g(x)dx
  • Growth / decay
Homogeneous
  • F(y/x)
  • y = vx
Linear
  • IF = e∫P
  • y·IF = ∫Q·IF
  • Bernoulli

8. Tips and Tricks

Shortcuts
  • Useful IFs: P = 1/x → IF = x; P = 2/x → x²; P = −1/x → 1/x; P = tan x → sec x; P = cot x → sin x.
  • Before choosing a method, check whether the DE is linear in x (swap roles) — this often makes it easy.
  • Look for exact differentials like x dy + y dx hidden inside the equation.
  • In MCQs, you can differentiate each option and check which satisfies the DE and the initial condition.
Mistakes students make
  • Forgetting to write the DE in the standard form y′ + Py = Q (coefficient of y′ must be 1) before finding P.
  • Stating a degree for a DE that contains sin(y′) or √ of a derivative without first clearing it.
  • Dropping the constant C and then being unable to use the initial condition.

9. Important One-liners

  • The order of a DE equals the number of arbitrary constants in its general solution.
  • The DE of all straight lines y = mx + c is y″ = 0.
  • The DE of all circles touching the x-axis at the origin has order 1.
  • If IF = e∫P dx and P = 1/x, the IF is x (for x > 0).
  • A homogeneous DE stays the same when x and y are replaced by λx and λy in f(x, y)/g(x, y).
  • Newton's law of cooling, radioactive decay and compound growth all lead to dy/dt = ky.

10. PYQ-pattern Questions with Solutions

These questions are original and modelled on concepts frequently tested in JEE Main.

Q1Easy

The order and degree of (d²y/dx²)³ + (dy/dx)⁴ + y = 0 are respectively:

  1. 2, 3
  2. 3, 2
  3. 2, 4
  4. 4, 2
Show solution

The highest derivative is d²y/dx² (order 2), and its power is 3 (degree 3).

Answer: (A)

Q2Numerical

If dy/dx + y = eˣ and y(0) = 1, find y(ln 2).

Show solution

IF = eˣ, so (y eˣ)′ = e²ˣ ⇒ y eˣ = e²ˣ/2 + C. With y(0) = 1: 1 = 1/2 + C ⇒ C = 1/2.

So y = (eˣ + e−x)/2. At x = ln 2: y = (2 + 1/2)/2 = 5/4 = 1.25.

Answer: 1.25

Q3Medium

The general solution of dy/dx = (x + y)/x (x > 0) is:

  1. y = x ln x + Cx
  2. y = ln x + C
  3. y = x² + Cx
  4. y = Cx ln x
Show solution

Homogeneous: put y = vx. Then v + x dv/dx = 1 + v ⇒ dv = dx/x ⇒ v = ln x + C.

So y = x ln x + Cx. (Alternatively it is linear: y′ − y/x = 1 with IF = 1/x.)

Answer: (A)

Q4Numerical

If dy/dx = ex−y and y(0) = 0, find y(1).

Show solution

Separate: ey dy = eˣ dx ⇒ ey = eˣ + C. At x = 0, y = 0: 1 = 1 + C ⇒ C = 0.

So ey = eˣ, i.e. y = x, and y(1) = 1.

Answer: 1

Q5Medium

The order of the differential equation whose general solution is y = a cos(x + b), where a and b are arbitrary constants, is:

  1. 1
  2. 2
  3. 3
  4. 0
Show solution

There are two independent arbitrary constants, a and b. Differentiating twice: y″ = −a cos(x + b) = −y, so the DE is y″ + y = 0, of order 2.

Answer: (B)

Q6Hard

If x (dy/dx) + 2y = x² for x > 0 and y(1) = 1, then y(2) equals:

  1. 19/16
  2. 5/4
  3. 17/16
  4. 1
Show solution

Standard form: y′ + (2/x)y = x. IF = e∫2/x dx = x².

(y x²)′ = x³ ⇒ y x² = x⁴/4 + C. With y(1) = 1: 1 = 1/4 + C ⇒ C = 3/4.

y = x²/4 + 3/(4x²). At x = 2: y = 1 + 3/16 = 19/16.

Answer: (A)

Q7Medium

A population grows at a rate proportional to its size and doubles in 10 years. The time it takes to triple is:

  1. 15 years
  2. 10 ln 3/ln 2 years
  3. 30 years
  4. 20 years
Show solution

P = P0ekt with e10k = 2 ⇒ k = (ln 2)/10.

For tripling: ekt = 3 ⇒ t = (ln 3)/k = 10 ln 3/ln 2 ≈ 15.85 years.

Answer: (B)

Q8Hard

The solution of x dy − y dx = x² dx with y(1) = 2 is:

  1. y = x² + x
  2. y = x² + 1
  3. y = 2x²
  4. y = x + 1
Show solution

Divide by x²: (x dy − y dx)/x² = dx ⇒ d(y/x) = dx ⇒ y/x = x + C.

At x = 1, y = 2: 2 = 1 + C ⇒ C = 1. So y = x² + x.

Answer: (A)

11. Quick Revision Summary

Remember before the exam
  • Order = highest derivative; degree = its power (only if polynomial in derivatives).
  • Separable → integrate; homogeneous → y = vx.
  • Linear y′ + Py = Q → IF = e∫P dx, y·IF = ∫Q·IF dx + C.
  • Check linearity in x as well; look for exact differentials.
  • Growth/decay: P = P0ekt.
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