Mathematics · JEE

Differential Equations Formula Sheet for JEE

36+ JEE formulas in this unit

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The Differential Equations JEE formula sheet lists 36+ important formulas for JEE Main and Advanced, including essential identities from Ordinary differential equations, their order and degree, The solution of differential equation by the method of separation of variables, Solution of a homogeneous and linear differential equation of the type dy/dx + p(x)y = q(x). Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Differential Equations. This unit-wise formula list covers 36+ exam-relevant results across Ordinary differential equations, their order and degree, The solution of differential equation by the method of separation of variables, Solution of a homogeneous and linear differential equation of the type dy/dx + p(x)y = q(x), organised by subtopic for quick last-minute revision.

JEE Formula Sheet

36 formulas across 3 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 19Important: 15Supplementary: 2

Ordinary differential equations, their order and degree

Definition of an ordinary differential equationEssential

F(x,y,y',y'',\dots,y^{(n)})=0

Variables: $x$ is the independent variable, $y$ is the dependent variable, and $y',y'',\dots,y^{(n)}$ are derivatives of $y$ with respect to $x$ .
Conditions: Only one independent variable is involved.
Where used in JEE: To identify whether a given relation is an ordinary differential equation and to classify it.

Order of a differential equationEssential

\text{Order} =

highest order derivative present in the differential equation

Variables: Highest derivative may be $y^{(n)}$ .
Conditions: Derivative must actually appear in the equation.
Where used in JEE: Classification of differential equations; direct theory questions.

Degree of a differential equationEssential

\text{Degree} =

power of the highest order derivative after the equation is made free from radicals and fractions in derivatives

Variables: Highest order derivative usually denoted by $y^{(n)}$ .
Conditions: Defined only when the differential equation can be expressed as a polynomial in derivatives.
Where used in JEE: Questions asking order and degree after simplification.

General solution of a differential equationImportant

\phi(x,y,C_1,C_2,\dots,C_n)=0

Variables: $C_1,C_2,\dots,C_n$ are arbitrary constants.
Conditions: For an $n$ -th order differential equation, general solution typically contains $n$ arbitrary constants.
Where used in JEE: Identifying complete/general solution.

Particular solutionImportant

\phi(x,y,c_1,c_2,\dots,c_n)=0

Variables: $c_1,c_2,\dots,c_n$ are specific values of the arbitrary constants.
Conditions: Obtained from the general solution using given initial or boundary conditions.
Where used in JEE: Initial value problems and application-based DE questions.

Singular solutionSupplementary

\text{A solution not obtainable from the general solution by any choice of arbitrary constants}

Conditions: Arises in some first-order differential equations but is not central in most basic JEE solving methods.
Where used in JEE: Conceptual classification questions.

Formation of differential equation from family with one arbitrary constantImportant

\text{Differentiate once and eliminate the constant}

Variables: If family is $\phi(x,y,C)=0$ , eliminate $C$ using the equation and its derivative.
Conditions: Family contains one arbitrary constant.
Where used in JEE: Questions on forming a DE from a given family of curves.

Formation of differential equation from family with $n$ arbitrary constantsImportant

\text{Differentiate }n\text{ times and eliminate }C_1,C_2,\dots,C_n

Variables: $C_1,C_2,\dots,C_n$ are arbitrary constants.
Conditions: Family must contain $n$ independent arbitrary constants.
Where used in JEE: Formation of differential equation from a given family.

The solution of differential equation by the method of separation of variables

Variable separable differential equationEssential

\dfrac{dy}{dx}=f(x)g(y)

Variables: $f(x)$ is a function of $x$ , $g(y)$ is a function of $y$ .
Conditions: Must be expressible as product of a function of $x$ and a function of $y$ .
Where used in JEE: Recognition of equations solvable by separation of variables.

Separation of variables standard formEssential

\dfrac{dy}{g(y)}=f(x)\,dx

Variables: $g(y)\neq 0$ in the region of solution.
Conditions: Obtained from $\dfrac{dy}{dx}=f(x)g(y)$ .
Where used in JEE: Main transformation step in solving separable differential equations.

General integral for separable equationsEssential

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

Variables: $C$ is the constant of integration.
Conditions: Applicable when variables can be separated.
Where used in JEE: Direct solving of first-order separable DEs.

Alternative separable formImportant

M(x)\,dx+N(y)\,dy=0

Variables: $M(x)$ depends only on $x$ , $N(y)$ depends only on $y$ .
Conditions: Equation must split into pure $x$ -part and pure $y$ -part.
Where used in JEE: Recognition and direct integration of separable equations.

Solution of alternative separable formImportant

\int M(x)\,dx+\int N(y)\,dy=C

Variables: $C$ is an arbitrary constant.
Conditions: Applicable to $M(x)dx+N(y)dy=0$ .
Where used in JEE: Direct integration after separation.

Separable equation with factorable denominatorImportant

\dfrac{dy}{dx}=\dfrac{f(x)}{g(y)}\Rightarrow g(y)\,dy=f(x)\,dx

Variables: $f(x)$ , $g(y)$ are single-variable functions.
Conditions: Equation should already be in directly separable quotient form.
Where used in JEE: Quick rearrangement in standard JEE problems.

Initial condition application for separable DEImportant

C\text{ is determined from }y(x_0)=y_0

Variables: $(x_0,y_0)$ is the given initial point.
Conditions: Initial point must lie in the domain of the obtained solution.
Where used in JEE: Initial value problems after integrating a separable equation.

Exponential growth or decay formEssential

\dfrac{dy}{dx}=ky\Rightarrow \dfrac{dy}{y}=k\,dx\Rightarrow y=Ce^{kx}

Variables: $k$ is a constant, $C$ is an arbitrary constant.
Conditions: Valid for $y\neq 0$ during separation; $y=0$ is also a solution.
Where used in JEE: Standard separable model; repeated in growth-decay type questions.

Power-law separable formImportant

\dfrac{dy}{dx}=f(x)y^n\Rightarrow y^{-n}dy=f(x)dx

Variables: $n$ is a real constant.
Conditions: For $n\neq 1$ , integrate power form directly; for $n=1$ , use logarithmic form.
Where used in JEE: Common first-order separable equations.

Solution for power-law separable form when $n\neq 1$Important

\int y^{-n}dy=\int f(x)dx\Rightarrow \frac{y^{1-n}}{1-n}=\int f(x)dx+C

Variables: $n\neq 1$ , $C$ is an arbitrary constant.
Conditions: Requires $n\neq 1$ .
Where used in JEE: Bernoulli-like separable special cases and direct integration problems.

Standard logarithmic integration resultImportant

\int \dfrac{dy}{y}=\log|y|+C

Conditions: $y\neq 0$ .
Where used in JEE: Used repeatedly in separable and linear homogeneous equations.

Standard reciprocal integration resultImportant

\int \dfrac{dx}{x}=\log|x|+C

Conditions: $x\neq 0$ .
Where used in JEE: Used in homogeneous substitution method and many separable equations.

Solution of a homogeneous and linear differential equation of the type dy/dx + p(x)y = q(x)

Homogeneous first-order differential equation formEssential

\dfrac{dy}{dx}=F\!\left(\dfrac{y}{x}\right)

Variables: $F$ is a function of the ratio $y/x$ .
Conditions: Equation must depend only on $y/x$ or equivalently on ratio of variables.
Where used in JEE: Recognition of first-order homogeneous equations.

Equivalent homogeneous differential formImportant

M(x,y)\,dx+N(x,y)\,dy=0

with

M,N

homogeneous functions of the same degree

Variables: $M$ and $N$ satisfy $M(tx,ty)=t^nM(x,y)$ , $N(tx,ty)=t^nN(x,y)$ .
Conditions: Both functions must be homogeneous of the same degree.
Where used in JEE: Checking whether a DE is homogeneous before substitution.

Substitution for homogeneous first-order DEEssential

y=vx

Variables: $v$ is a function of $x$ .
Conditions: Used when DE is homogeneous in $y/x$ .
Where used in JEE: Standard substitution to reduce a homogeneous DE to separable form.

Derivative under substitution $y=vx$Essential

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

Variables: $v=v(x)$ .
Conditions: Follows from product rule after setting $y=vx$ .
Where used in JEE: Reduction of homogeneous equations to separable form.

Reduction of homogeneous equation to separable formEssential

v+x\dfrac{dv}{dx}=F(v)\Rightarrow x\dfrac{dv}{dx}=F(v)-v\Rightarrow \dfrac{dv}{F(v)-v}=\dfrac{dx}{x}

Variables: $v=y/x$ .
Conditions: Applicable after substituting $y=vx$ in $\dfrac{dy}{dx}=F(y/x)$ .
Where used in JEE: Main solving step for homogeneous first-order equations.

General solution pattern of homogeneous first-order DEEssential

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

Variables: $v=y/x$ , $C$ is an arbitrary constant.
Conditions: After substitution $y=vx$ .
Where used in JEE: Direct integration in homogeneous DE problems.

Linear differential equation of first orderEssential

\dfrac{dy}{dx}+p(x)y=q(x)

Variables: $p(x)$ and $q(x)$ are functions of $x$ .
Conditions: First-order and first-degree in $y$ .
Where used in JEE: Recognition of standard linear DE solvable by integrating factor.

Integrating factor for linear DEEssential

\mathrm{I.F.}=e^{\int p(x)\,dx}

Variables: $p(x)$ is the coefficient of $y$ in $\dfrac{dy}{dx}+p(x)y=q(x)$ .
Conditions: Applicable to first-order linear equations in standard form.
Where used in JEE: Core formula for solving linear differential equations.

Product derivative after multiplying by integrating factorEssential

\dfrac{d}{dx}\left[y\,e^{\int p(x)dx}\right]=q(x)e^{\int p(x)dx}

Variables: $y$ is the dependent variable, $p(x),q(x)$ are given functions.
Conditions: Equation must be in the form $\dfrac{dy}{dx}+p(x)y=q(x)$ .
Where used in JEE: Key transformation step in integrating factor method.

General solution of linear first-order DEEssential

y\,e^{\int p(x)dx}=\int q(x)e^{\int p(x)dx}\,dx + C

Variables: $C$ is an arbitrary constant.
Conditions: Applicable to $\dfrac{dy}{dx}+p(x)y=q(x)$ .
Where used in JEE: Final standard result for solving linear equations by integrating factor.

Explicit solution of linear first-order DEEssential

y=e^{-\int p(x)dx}\left(\int q(x)e^{\int p(x)dx}\,dx + C\right)

Variables: $C$ is an arbitrary constant.
Conditions: Equivalent explicit form of the standard linear solution.
Where used in JEE: Writing final answer directly in solved form for $y$.

Linear homogeneous first-order DEEssential

\dfrac{dy}{dx}+p(x)y=0

Variables: $p(x)$ is a function of $x$ .
Conditions: This is the special case $q(x)=0$ .
Where used in JEE: Special linear homogeneous equations.

Solution of linear homogeneous first-order DEEssential

y=Ce^{-\int p(x)dx}

Variables: $C$ is an arbitrary constant.
Conditions: Obtained from either separation or integrating factor.
Where used in JEE: Frequently tested direct result for homogeneous linear first-order equations.

Alternative derivation of linear homogeneous first-order DEImportant

\dfrac{dy}{dx}=-p(x)y\Rightarrow \dfrac{dy}{y}=-p(x)dx\Rightarrow \log|y|=-\int p(x)dx + C

Variables: $p(x)$ is a function of $x$ .
Conditions: Valid for $y\neq 0$ during separation; $y=0$ also satisfies the equation.
Where used in JEE: Quick solution of the homogeneous linear case by separation.

Initial value application for linear DEImportant

C\text{ is determined from }y(x_0)=y_0

Variables: $(x_0,y_0)$ is the given initial point.
Conditions: Initial point must satisfy domain restrictions of the coefficient functions and solution.
Where used in JEE: Initial value problems involving linear first-order equations.

Condition for exact recognition that a linear DE is already in derivative formSupplementary

\dfrac{dy}{dx}+p(x)y=q(x)\iff \dfrac{d}{dx}[\mu(x)y]=\mu(x)q(x)

when

\mu'(x)=p(x)\mu(x)

Variables: $\mu(x)$ is the integrating factor.
Conditions: $\mu(x)\neq 0$ .
Where used in JEE: Transforming a linear DE into exact derivative form.

Frequently asked questions

What are the important Differential Equations formulas for JEE?

This page lists 36+ JEE-relevant Differential Equations formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Differential Equations formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Differential Equations, covering Ordinary differential equations, their order and degree, The solution of differential equation by the method of separation of variables, Solution of a homogeneous and linear differential equation of the type dy/dx + p(x)y = q(x).

How should I revise the Differential Equations formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Differential Equations MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Differential Equations?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.