Differential Equations — JEE Mains Maths Formulas and Traps

Forming a Differential Equation

Eliminating the constants of a family

Order = number of constants

y=f(x;c_1,\dots,c_n)\ \Rightarrow\ F\big(x,y,y',\dots,y^{(n)}\big)=0

Order, degree, and the equation a solution satisfies

Degree after clearing radicals

y=x y'+\sqrt{1+y'^2}\ \Rightarrow\ (y-xy')^2=1+y'^2\ :\ \text{order }1,\ \text{degree }2

Common traps

Use the conditions first

A general circle has three constants. Conditions such as 'through the origin' or 'centre on

y=x

' remove some of them before you differentiate; skipping that step gives an equation of too high an order.

Clear the radical before counting

y=xy'+\sqrt{1+y'^2}

looks like degree 1, but squaring to remove the root gives

y'^2

terms, so the degree is 2. The degree is read only after the radical is gone.

Separating the Variables

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Separate, integrate, fix the constant

Separable form

\frac{dy}{dx}=f(x)\,g(y)\ \Rightarrow\ \int\frac{dy}{g(y)}=\int f(x)\,dx+C

When the slope depends on x alone

Direct integration

\frac{dy}{dx}=f(x)\ \Rightarrow\ y=\int f(x)\,dx+C

Substituting for x + y, and exact differentials

Substitution for a linear combination

t=ax+by+c\ \Rightarrow\ \frac{dt}{dx}=a+b\,\frac{dy}{dx}

Common traps

Constant before exponentiating

From

\ln y=x^2+C

, the solution is

y=Ae^{x^2}

, not

e^{x^2}+C

. Fix the constant in whichever form you keep, and never add it after exponentiating.

Two conditions for a second-order equation

Integrating twice brings two constants, so it needs two conditions — often one on the derivative and one on the value.

Differentiate the substitution fully

With

t=2x+3y

,

\frac{dt}{dx}=2+3\frac{dy}{dx}

, not

3\frac{dy}{dx}

. Dropping the 2 gives a separable equation with the wrong answer.

Homogeneous Equations

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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}

Common traps

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.

Parallel lines need a different substitution

If

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.

Linear Equations: The Integrating Factor

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Standard form and the integrating factor

Linear equation

\frac{dy}{dx}+Py=Q\ \Rightarrow\ y\,e^{\int P\,dx}=\int Q\,e^{\int P\,dx}\,dx+C

Integrating factors from trigonometric P

Common trigonometric factors

P=k\tan x\ \Rightarrow\ \mu=\sec^kx,\qquad P=\frac{1}{1+x^2}\ \Rightarrow\ \mu=e^{\tan^{-1}x}

Common traps

Standard form before the factor

The integrating factor comes from

P

only after the coefficient of

y'

is 1. In

x\,y'+2y=x^2

,

P

is

\frac2x

, not 2.

Sign of the tangent term

\int\tan x\,dx=\ln\sec x

, so

+\tan x

gives

\sec x

and

-\tan x

gives

\cos x

. Swapping them is the commonest slip on this page.

Integrating Factors in Disguise

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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)

Common traps

Divide only if it helps

Dividing

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

by

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.

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.

Equations Reducible to Linear

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Linear in x: dx/dy + P(y)x = Q(y)

Linear in x

\frac{dx}{dy}+P(y)\,x=Q(y)\ \Rightarrow\ x\,e^{\int P\,dy}=\int Q\,e^{\int P\,dy}\,dy+C

Bernoulli equations

Bernoulli substitution

y'+Py=Qy^n,\ z=y^{1-n}\ \Rightarrow\ z'+(1-n)P\,z=(1-n)Q

Substituting for a function of y

Substitution

f'(y)\,\frac{dy}{dx}+P(x)\,f(y)=Q(x),\ u=f(y)\ \Rightarrow\ \frac{du}{dx}+Pu=Q

Common traps

Integrate in y throughout

Once

x

is the unknown, every integral is with respect to

y

, including the integrating factor. Writing

e^{\int P\,dx}

out of habit mixes the variables.

The factor 1 - n

With

z=y^{-1}

,

z'=-y^{-2}y'

: the minus sign flips both

P

and

Q

. Forgetting it gives an integrating factor of the wrong sign.

sin 2y becomes 2 tan y

After dividing by

\cos^2y

,

\sin2y

becomes

2\tan y

, not

\tan y

. The factor 2 goes straight into the integrating factor.

Using a Linear Solution

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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

Common traps

Which infinity

e^{\alpha x}\to0

as

x\to-\infty

only when

\alpha>0

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

\alpha

the condition forces.

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.

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.

Equations Hidden in Integrals and Limits

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Differentiating an integral equation

Fundamental theorem

f(x)=h(x)+\int_a^xg\big(t,f(t)\big)\,dt\ \Rightarrow\ f'(x)=h'(x)+g\big(x,f(x)\big),\ f(a)=h(a)

Equations from limits and derivative rules

The limit as a derivative

\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=2xf(x)-x^2f'(x)

Common traps

The starting value is free

Differentiating throws away the constant, so the equation alone cannot give

f

. Put the lower limit into the original relation to get

f(a)

— the question almost never states it.

Watch the order in the denominator

\frac{\dots}{x-t}

and

\frac{\dots}{t-x}

differ by a sign. Differentiate in

t

and divide by the derivative of the denominator in

t

, which is

-1

for

x-t

.

Curves and Growth from Rates

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Curves from tangent, normal and area conditions

Tangent at (x, y)

Y-y=\frac{dy}{dx}(X-x):\quad X\text{-int}=x-\frac{y}{y'},\ \ Y\text{-int}=y-xy'

Growth, decay and cooling

Newton's law of cooling

\frac{dT}{dt}=-k(T-A)\ \Rightarrow\ T-A=(T_0-A)\,e^{-kt}

Common traps

Midpoint and intercept are not the same

'The

y

-axis bisects

PQ

' means the midpoint has

x=0

, i.e.

x+\left(x-\frac{y}{y'}\right)=0

. Setting the intercept itself to zero is a different condition.

The difference decays, not the temperature

In cooling,

T-A

is multiplied by

e^{-kt}

, not

T

itself. Halving the temperature instead of its excess over the room gives a wrong answer.

Differential Equations formulas

Forming a Differential Equation

Eliminating the constants of a family

Order, degree, and the equation a solution satisfies

Common traps

Separating the Variables

Separate, integrate, fix the constant

When the slope depends on x alone

Substituting for x + y, and exact differentials

Common traps

Homogeneous Equations

Putting y = vx

When y = vx is not the first move

Common traps

Linear Equations: The Integrating Factor

Standard form and the integrating factor

Integrating factors from trigonometric P

Common traps

Integrating Factors in Disguise

The left side is already a derivative

When P is the derivative of a logarithm

Common traps

Equations Reducible to Linear

Linear in x: dx/dy + P(y)x = Q(y)

Bernoulli equations

Substituting for a function of y

Common traps

Using a Linear Solution

Limits that fix the constant

Maxima, critical points and derivatives of the solution

Integrating the solution: odd parts and areas

Common traps

Equations Hidden in Integrals and Limits

Differentiating an integral equation

Equations from limits and derivative rules

Common traps

Curves and Growth from Rates

Curves from tangent, normal and area conditions

Growth, decay and cooling

Common traps

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