JEE Mains Maths · Differential Equations

Using a Linear Solution

Questions where solving the linear equation is only the first half: the constant is fixed by a limit instead of a point, or the solution is then maximised, differentiated, integrated or compared with another solution.

All Differential Equations notes JEE Mains Maths strategy

Why this matters

Twenty-two PYQs, eight of them numerical answer, and 2022 alone has nine. The equation is routine; the marks are in what comes after. Seven fix the constant from behaviour at infinity or compare two solutions, eight need a maximum or a derivative of the solution, and seven integrate it. Three ideas cover the page.

Concept 1 of 3: Limits that fix the constant

The general solution of

y'+ky=Q

is a particular part plus

Ce^{-kx}

. If the question says the limit at

+\infty

-\infty

is finite, the exponential part must vanish there or be killed: either

C=0

, or the exponent's sign is forced. Two solutions of the same linear equation differ by

Ce^{-\int P}

, which is never zero unless

C=0

, so distinct solutions never cross.

Definition

$y=y_p+Ce^{-\int P\,dx}$ .
$\lim_{x\to\infty}y$ finite and $e^{x}$ in the solution: $C=0$ .
$f'=\alpha f+\beta$ : $f=Ce^{\alpha x}-\frac\beta\alpha$ ; a finite limit as $x\to-\infty$ needs $\alpha>0$ and equals $-\frac\beta\alpha$ .
Two distinct solutions of a linear equation never intersect.

General solution with constant coefficient

y'+ky=Q\ \Rightarrow\ y=y_p(x)+Ce^{-kx}

Worked example

f'(x)=2f(x)-4

and

\lim_{x\to-\infty}f(x)

is finite, with

f(0)=3

. Find

f(\ln2)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 1 February 2024 · Q157Moderate

Example 1 · Differential Equations · Using a Linear Solution

Let

\alpha

be a non-zero real number. Suppose

f:R \rightarrow

R

is a differentiable function such that

f(0) = 2

and

\lim_{x \rightarrow - \infty} f(x) = 1

. If

f^{'}(x) = \alpha f(x) + 3

, for all

x \in R

, then

f\left( -\log_{e}2 \right)

is equal to

Which infinity

e^{\alpha x}\to0

x\to-\infty

only when

\alpha>0

. Read which end the limit is taken at before deciding which sign of

\alpha

the condition forces.

Drill 6 more on limits that fix the constant

Concept 2 of 3: Maxima, critical points and derivatives of the solution

For an extremum, you rarely need the solution's formula differentiated from scratch: the equation itself gives

y'

. Set

y'=0

using the equation, or write the solution in a simple variable such as

t=\cos x

and maximise the quadratic. For a combination like

xy''+2y'

, notice it is

(xy)''

Definition

Critical point: $y'=0$ , read from the equation.
$y=\cos x-2\cos^2x$ : a quadratic in $t=\cos x$ , maximum at $t=\frac14$ .
$(xy)'=xy'+y$ , $(xy)''=xy''+2y'$ .
$z=x^2y-e^x$ with $xy'+2y=xe^x$ : $z'=x(xy'+2y)-e^x$ .

Derivative from the equation

\frac{dy}{dx}=Q(x)-P(x)\,y\ \ \text{(no need to differentiate the solution)}

Worked example

y'+y\tan x=\sin x

with

y(0)=0

. Find the maximum of

y

\left(0,\frac\pi2\right)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 26 July 2022 · Q71Moderate

Example 2 · Differential Equations · Using a Linear Solution

\frac{dy}{dx}+ 2ytanx = \sin x,0 < x <\frac{\pi}{2}

and

y\left( \frac{\pi}{3} \right)=

0, then the maximum value of

y(x)

Check the endpoint and the domain

A quadratic in

t=\cos x

has its vertex inside the range only if that value of

t

is attainable on the given interval. Otherwise the extreme value is at an end.

Drill 7 more on maxima, critical points and derivatives of the solution

Concept 3 of 3: Integrating the solution: odd parts and areas

Many questions ask for

\int_{-a}^{a}y\,dx

after solving. Split the solution into odd and even parts: the odd part integrates to zero on a symmetric interval, so only the even part needs work. For an area between the solution and a line, find where they meet and integrate the difference.

Definition

$\int_{-a}^{a}g=0$ for odd $g$ ; $=2\int_0^ag$ for even $g$ .
Typical: $y\sqrt{1-x^2}=\frac{x^5}{5}+x^2$ : the $x^5$ term is odd.
$\int\frac{x^2}{\sqrt{1-x^2}}dx$ : put $x=\sin\theta$ .
Area between $y=f$ and a line: $\int|f-\text{line}|$ between the meeting points.

Symmetric interval

\int_{-a}^{a}\big(g_{\text{odd}}+g_{\text{even}}\big)\,dx=2\int_0^{a}g_{\text{even}}\,dx

Worked example

y=x^3+x^2\cos x

solves an equation. Find

\int_{-1}^{1}y\,dx

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 26 July 2022 · Q160Moderate

Example 3 · Differential Equations · Using a Linear Solution

Let the solution curve

y = f(x)

of the differential equation

\frac{dy}{dx}+\frac{xy}{x^{2}- 1}=\frac{x^{4}+ 2x}{\sqrt{1 -x^{2}}},x \in ( - 1,1)

pass through the origin. Then

\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x)dx

is equal to

Split before integrating

Integrating the whole solution term by term on

[-a,a]

wastes time on parts that cancel. Identify the odd terms first and drop them.

Drill 6 more on integrating the solution: odd parts and areas

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 (3)

Limits that fix the constant
General solution with constant coefficient
$y'+ky=Q\ \Rightarrow\ y=y_p(x)+Ce^{-kx}$
Maxima, critical points and derivatives of the solution
Derivative from the equation
$\frac{dy}{dx}=Q(x)-P(x)\,y\ \ \text{(no need to differentiate the solution)}$
Integrating the solution: odd parts and areas
Symmetric interval
$\int_{-a}^{a}\big(g_{\text{odd}}+g_{\text{even}}\big)\,dx=2\int_0^{a}g_{\text{even}}\,dx$

Watch out for (3)

Which infinity
→ Limits that fix the constant
Check the endpoint and the domain
→ Maxima, critical points and derivatives of the solution
Split before integrating
→ Integrating the solution: odd parts and areas

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