JEE Mains Maths · Differential Equations

Integrating Factors in Disguise

Linear equations where the integrating factor is not found by a standard integral: the left side is already the derivative of a product, or P is the derivative of the logarithm of some function.

All Differential Equations notes JEE Mains Maths strategy

Why this matters

Twenty-two PYQs, and they look much harder than they are. A long coefficient of y is usually the derivative of a log, so its integrating factor is a simple quotient; and a messy left side is often already d(uy)/dx. Spotting that is the whole question. Two ideas cover the page.

Concept 1 of 2: The left side is already a derivative

Before computing anything, check whether the left side is the derivative of a product:

u\,y'+u'\,y=(uy)'

. If the coefficient of

y

is the derivative of the coefficient of

y'

, integrate both sides at once. Likewise

u\,y'-u'\,y

over

u^2

\left(\frac yu\right)'

. Simplifying a heavy coefficient first — or moving the origin — often reveals the same pattern.

Definition

$u\,y'+u'\,y=(uy)'$ : e.g. $x^4y'+4x^3y=(x^4y)'$ .
$\frac{u\,y'-u'\,y}{u^2}=\left(\frac yu\right)'$ .
$(1+\cos^2x)'=-\sin2x$ , $(x^2+4)'=2x$ .
Simplify first: $\sin x(\sec x-\sin x\tan x)=\sin x\cos x$ .

Product rule in reverse

u\,\frac{dy}{dx}+\frac{du}{dx}\,y=\frac{d}{dx}(uy)

Worked example

Solve

(1+x^2)\frac{dy}{dx}+2xy=\cos x

with

y(0)=0

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q168Moderate

Example 1 · Differential Equations · Integrating Factors in Disguise

Let

y = y(x)

be the solution of the differential equation

\left( x^{2}+ 4 \right)^{2}dy +\left( 2x^{3}y + 8xy - 2 \right)dx = 0

. If

y(0) = 0

then

y(2)

is equal to

Divide only if it helps

Dividing

x^4y'+4x^3y=f

x^4

and computing

e^{\int4/x}=x^4

just multiplies back by

x^4

. If the left side is already exact, integrate it directly.

Drill 8 more on the left side is already a derivative

Concept 2 of 2: When P is the derivative of a logarithm

P=\frac{g'}{g}

, then

\int P\,dx=\ln g

and the integrating factor is simply

g

. A long rational or exponential coefficient is usually built this way: test the derivative of the denominator, or split by partial fractions into pieces like

\frac1x

\frac{2}{x+1}

-\frac{1}{x+3}

. The right side is then chosen so that

\mu Q

integrates cleanly.

Definition

$P=\frac{g'}{g}\Rightarrow\mu=g$ ; $P=\frac{g'}{g}-\frac{h'}{h}\Rightarrow\mu=\frac gh$ .
Partial fractions: $\frac{5}{x(x^5+1)}=\frac5x-\frac{5x^4}{x^5+1}$ , so $\mu=\frac{x^5}{x^5+1}$ .
$P=u'$ for a known $u$ : $\mu=e^{u}$ , and often $Q=e^{u}\cdot(\dots)$ .
$\int\frac{dx}{x^2-1}=\frac12\ln\frac{x-1}{x+1}$ .

Log-derivative coefficient

P=\frac{g'(x)}{g(x)}\ \Rightarrow\ e^{\int P\,dx}=g(x)

Worked example

Find the integrating factor of

\frac{dy}{dx}+\frac{3x^2+1}{x^3+x}\,y=1

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 July 2022 · Q160Moderate

Example 2 · Differential Equations · Integrating Factors in Disguise

Let

y = y(x)

be the solution curve of the differential equation

\frac{dy}{dx}+\left( \frac{2x^{2}+ 11x + 13}{x^{3}+ 6x^{2}+ 11x + 6} \right)

y =\frac{(x + 3)}{x + 1},x > - 1

, which passes through the point

(0,1)

. Then

y(1)

is equal to:

Check the derivative, do not integrate blindly

Differentiate the denominator and compare with the numerator before attempting the integral. When they match, the factor is the denominator itself; attempting a long integral wastes the time the question is testing.

Drill 12 more on when p is the derivative of a logarithm

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The left side is already a derivative
Product rule in reverse
$u\,\frac{dy}{dx}+\frac{du}{dx}\,y=\frac{d}{dx}(uy)$
When P is the derivative of a logarithm
Log-derivative coefficient
$P=\frac{g'(x)}{g(x)}\ \Rightarrow\ e^{\int P\,dx}=g(x)$

Watch out for (2)

Divide only if it helps
→ The left side is already a derivative
Check the derivative, do not integrate blindly
→ When P is the derivative of a logarithm

Test yourself on Differential Equations

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Differential Equations chapter test Drill all 22 questions