Differential Equation formulas

Every formula JEE Advanced expects you to know for Differential Equation, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 32
Subject: Mathematics
Exam: JEE Advanced
No sign-in needed: ₹0

Differential Equation, every formula

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Order of a DE
$order = order of the highest derivative present$
Degree of a DE
$degree = power of the highest-order derivative (rational, radical-free form)$
General solution
$contains as many arbitrary constants as the order$
Formation of a DE
$eliminate n arbitrary constants \Rightarrow DE of order n$
Variable separable
$\frac{d y}{d x} = \frac{f ( x )}{g ( y )} \Rightarrow \int g (y) d y = \int f (x) d x$
Reducible to separable
$\frac{d y}{d x} = f (a x + b y + c) : put z = a x + b y + c$
Homogeneous function
$f (k x, k y) = k^{n} f (x, y)$
degree n
Homogeneous DE substitution
$\frac{d y}{d x} = F (\frac{y}{x}) : put y = v x, \frac{d y}{d x} = v + x \frac{d v}{d x}$
Homogeneous DE (x=vy)
$put x = v y, \frac{d x}{d y} = v + y \frac{d v}{d y}$
Non-homogeneous (parallel lines)
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} : put a_{1} x + b_{1} y = t$
Non-homogeneous (shift origin)
$put x = X + h, y = Y + k with a_{i} h + b_{i} k + c_{i} = 0$
Linear DE in y
$\frac{d y}{d x} + P (x) y = Q (x)$
Integrating factor (in y)
$I.F. = e^{\int P d x}$
Linear DE solution
$y \cdot (I.F.) = \int Q \cdot (I.F.) d x + C$
Linear DE in x
$\frac{d x}{d y} + P (y) x = Q (y), I.F. = e^{\int P d y}$
Bernoulli's equation
$\frac{d y}{d x} + P (x) y = Q (x) y^{n}$
divide by y^n, put z=y^{1-n}
Bernoulli substitution
$z = y^{1 - n} \Rightarrow linear in z$
Orthogonal trajectory rule
$replace \frac{d y}{d x} \to - \frac{d x}{d y}$
Exact differential xdy+ydx
$x d y + y d x = d (x y)$
Exact differential (ydx-xdy)/y²
$\frac{y d x - x d y}{y ^{2}} = d (\frac{x}{y})$
Exact differential (xdy-ydx)/x²
$\frac{x d y - y d x}{x ^{2}} = d (\frac{y}{x})$
d(x/y)
$d (\frac{x}{y}) = \frac{2 x y d x - x ^{2} d y}{y ^{2}} (for x^{2} / y)$
as in the table
tan⁻¹(y/x) differential
$\frac{x d y - y d x}{x ^{2} + y ^{2}} = d (tan^{- 1} \frac{y}{x})$
log(xy) differential
$\frac{x d y + y d x}{x y} = d (ln x y)$
log(y/x) differential
$\frac{x d y - y d x}{x y} = d (ln \frac{y}{x})$
d(x²+y²)
$2 (x d x + y d y) = d (x^{2} + y^{2})$
y=A e^{αx}+B e^{βx} ⇒ DE
$y_{2} - (α + β) y_{1} + α β y = 0$
y=e^{αx}(A cosβx+B sinβx) ⇒ DE
$y_{2} - 2 α y_{1} + (α^{2} + β^{2}) y = 0$
Tangent intercept on x-axis
$x - \frac{y}{d y / d x}$
Tangent intercept on y-axis
$y - x \frac{d y}{d x}$
dy/dx = e^{x} solution
$y = e^{x} + C$
simple separable
Family y=mx orthogonal to circles
$y = m x ⊥ x^{2} + y^{2} = a^{2}$

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Differential Equation formulas

Differential Equation, every formula

Order of a DE

Degree of a DE

General solution

Formation of a DE

Variable separable

Reducible to separable

Homogeneous function

Homogeneous DE substitution

Homogeneous DE (x=vy)

Non-homogeneous (parallel lines)

Non-homogeneous (shift origin)

Linear DE in y

Integrating factor (in y)

Linear DE solution

Linear DE in x

Bernoulli's equation

Bernoulli substitution

Orthogonal trajectory rule

Exact differential xdy+ydx

Exact differential (ydx-xdy)/y²

Exact differential (xdy-ydx)/x²

d(x/y)

tan⁻¹(y/x) differential

log(xy) differential

log(y/x) differential

d(x²+y²)

y=A e^{αx}+B e^{βx} ⇒ DE

y=e^{αx}(A cosβx+B sinβx) ⇒ DE

Tangent intercept on x-axis

Tangent intercept on y-axis

dy/dx = e^{x} solution

Family y=mx orthogonal to circles

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