JEE Mains Maths · Differential Equations

Homogeneous Equations

Equations where the slope depends only on the ratio y/x, solved by putting y = vx; and the variants that need x = vy, a shift of origin, or a power substitution first.

All Differential Equations notes JEE Mains Maths strategy

Why this matters

Twenty-four PYQs, twenty of them multiple choice. Most are the standard move: y = vx turns the equation into a separable one in v and x. The rest hide the homogeneous form behind x as the dependent variable, a constant term that a shift of origin removes, or y² where y should be. Two ideas cover the page.

Concept 1 of 2: Putting y = vx

An equation is homogeneous when every term has the same total degree in

x

and

y

, so the slope can be written as

F\left(\frac yx\right)

. Put

y=vx

: then

\frac{dy}{dx}=v+x\frac{dv}{dx}

, and the equation becomes

x\frac{dv}{dx}=F(v)-v

, which separates. Return to

y

at the end, then use the given point.

Definition

Test: replacing $x,y$ by $tx,ty$ leaves the slope unchanged.
$y=vx\Rightarrow y'=v+xv'$ and $\frac{dv}{F(v)-v}=\frac{dx}{x}$ .
$x\,dy-y\,dx=x^2\,dv$ when $y=vx$ .
$\int\frac{dv}{\sqrt{1+v^2}}=\ln\left(v+\sqrt{1+v^2}\right)$ , $\int\frac{dv}{\sqrt{1-v^2}}=\sin^{-1}v$ .

The substitution

y=vx:\quad v+x\frac{dv}{dx}=F(v)\ \Rightarrow\ \int\frac{dv}{F(v)-v}=\ln|x|+C

Worked example

Solve

\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}

with

y(1)=1

, and find

y(e)^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 June 2022 · Q62Moderate

Example 1 · Differential Equations · Homogeneous Equations

Let the solution curve of the differential equation

x\frac{dy}{dx}- y =\sqrt{y^{2}+ 16x^{2}},y(1) = 3

y = y(x)

. Then

y(2)

is equal to:

Return to y before using the point

The condition is given in

x

and

y

. Either convert it to

v=\frac yx

at that point, or substitute back first; mixing the two gives the wrong constant.

Drill 16 more on putting y = vx

Concept 2 of 2: When y = vx is not the first move

Three variants reach the same method. If the equation is simpler with

x

as the dependent variable, put

x=vy

. If numerator and denominator are linear with constant terms, move the origin to where both lines meet,

x=X+h

y=Y+k

, and the constants vanish. If

y

appears only as

y^2

(with

y\,dy

), put

t=y^2

and the equation becomes homogeneous in

x

and

t

Definition

$\frac{dx}{dy}=F\left(\frac xy\right)$ : put $x=vy$ , $\frac{dx}{dy}=v+y\frac{dv}{dy}$ .
$\frac{dy}{dx}=\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}$ : solve $a_1h+b_1k+c_1=0$ , $a_2h+b_2k+c_2=0$ and shift.
$y\,dy$ with $y^2$ elsewhere: put $t=y^2$ , $dt=2y\,dy$ .

Shift of origin

x=X+h,\ y=Y+k:\quad \frac{dY}{dX}=\frac{a_1X+b_1Y}{a_2X+b_2Y}

Worked example

Reduce

\frac{dy}{dx}=\frac{x+y-4}{x-y+2}

to a homogeneous equation.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 July 2022 · Q159Moderate

Example 2 · Differential Equations · Homogeneous Equations

If the solution curve of the differential equation

\frac{dy}{dx}=\frac{x + y - 2}{x - y}

passes through the point

(2,1)

and

(k + 1,2),k > 0

, then

Parallel lines need a different substitution

a_1x+b_1y

and

a_2x+b_2y

are proportional, the two lines never meet and no shift exists. Put

t=a_1x+b_1y

instead; the equation then separates.

Drill 6 more on when y = vx is not the first move

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)

Putting y = vx
The substitution
$y=vx:\quad v+x\frac{dv}{dx}=F(v)\ \Rightarrow\ \int\frac{dv}{F(v)-v}=\ln|x|+C$
When y = vx is not the first move
Shift of origin
$x=X+h,\ y=Y+k:\quad \frac{dY}{dX}=\frac{a_1X+b_1Y}{a_2X+b_2Y}$

Watch out for (2)

Return to y before using the point
→ Putting y = vx
Parallel lines need a different substitution
→ When y = vx is not the first move

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 24 questions