JEE · Mathematics

Differential Equations

Classify and solve the first-order ordinary differential equations in current JEE scope, preserve valid constant solutions before division, use separation/homogeneous/linear methods under their conditions, and verify solutions against the original equation.

Subject
Mathematics
Syllabus unit
Differential Equations
  • Explicit in both JEE Main 2026 Unit 9 and JEE Advanced 2026 Integral Calculus
  • Formation of differential equations is Advanced-explicit, not separately named in current Main Unit 9
  • No invented weightage, question counts or trend percentages

Content status: draft. Verified academic content for this page has not been loaded yet, so the page is excluded from search indexing and the sitemap.

In short

A differential equation relates an unknown function to one or more of its derivatives. JEE solving begins by classifying the equation before manipulating it.

For separable equations, check constant solutions before dividing by a function of y. For a linear first-order equation y'+P(x)y=Q(x), use an integrating factor on an interval where the coefficients are valid. Verify the final family in the original equation.

Syllabus mapping

  • Unit
    Differential Equations
    Topics
    Order and degree, Separation of variables, Homogeneous differential equations, Linear first-order differential equations, Formation of ordinary differential equations

What this chapter contains and why it matters

  • Question
    What is the chapter about?
    Direct answer
    Classifying and solving first-order ordinary differential equations by separable, homogeneous, and linear methods, with constant-solution and interval checks.
  • Question
    What is the central method choice?
    Direct answer
    Classify the equation as separable, homogeneous, or linear before choosing a solution technique.
  • Question
    Where do most mistakes begin?
    Direct answer
    Assigning a degree when the equation is not polynomial in derivatives, dividing by y and losing a constant solution, and presenting formation as explicit Main Unit 9 scope.
  • Question
    What should come before Differential Equations?
    Direct answer
    Differentiation and Integration, for elimination of constants and solving separated/transformed equations.
  • Question
    What comes after it?
    Direct answer
    Onward practice through previous-year papers, where route state and provenance permit.

Official JEE syllabus documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Differential Equations

Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.

  • Concept group
    Order, degree, separation, homogeneous, linear
    JEE Main 2026
    Explicitly includes ordinary differential equations, order and degree, separation of variables, homogeneous differential equations and linear differential equations of the stated first-order form dy/dx+p(x)y=q(x).
    JEE Advanced 2026
    Explicitly includes homogeneous differential equations of first order and first degree, separation of variables and linear first-order differential equations.
    Preparation note
    Both papers share the same three solving methods; verify the exact stated form before applying the linear method.
  • Concept group
    Formation of differential equations
    JEE Main 2026
    Not separately named in the current Main Unit 9 wording.
    JEE Advanced 2026
    Explicitly includes formation of ordinary differential equations.
    Preparation note
    Treat formation as Advanced-explicit only; do not present it as current Main-explicit scope.

Sources: official Main Unit 9 and official Advanced Integral Calculus differential-equation scope, linked in the sources section below.

Before this chapter

Prerequisites: what you should know before Differential Equations

  • Prerequisite
    Differentiation
    You are ready if you can…
    Differentiate a solution family and eliminate arbitrary constants.
    If not, repair this first
    Revise Differentiation.
  • Prerequisite
    Indefinite integration
    You are ready if you can…
    Integrate a separated equation on each side.
    If not, repair this first
    Revise Integration.
  • Prerequisite
    Logarithms and exponentials
    You are ready if you can…
    Convert ln|y|=x²+C into an exponential form.
    If not, repair this first
    Revise logarithm and exponential properties.
  • Prerequisite
    Algebraic factorisation
    You are ready if you can…
    Recognise when a right-hand side factors as g(x)h(y).
    If not, repair this first
    Revise factorisation.

This is a readiness check, not a weightage or scoring-priority list.

Concepts in this chapter

1. Determine order and degree only when degree is defined

Degree is undefined when the equation is not polynomial in its derivative symbols.

Determine the order and degree of the equation; degree is defined only after the equation is polynomial in its derivative symbols, otherwise it is undefined.

2. Identify separable, homogeneous or linear structure

Correct classification decides which method applies.

Identify whether the equation is separable, homogeneous of the standard school form, or first-order linear, before selecting a solution method.

3. Preserve constant solutions before dividing

Dividing by h(y) can silently discard a valid constant solution.

Before dividing by h(y) in a separable equation, solve h(y)=0 for possible constant solutions.

4. Restrict to intervals where a coefficient is nonzero

Division by an x-dependent coefficient requires interval control.

Before dividing by an x-dependent coefficient, restrict the working interval to where it is nonzero.

5. Integrate, include the constant, and apply any given condition

An initial condition must lie within the solution's valid domain.

Integrate the separated or transformed equation, include the constant, and apply an initial condition if supplied, checking it lies in the working interval.

6. Differentiate the result and check it in the original equation

A solved family must be verified against the original differential equation.

Differentiate the final solution and substitute back into the original equation to confirm it is satisfied.

Method selector for first-order differential equations

Match the equation signal to the correct first method.

  • Equation signal
    Variables already factor by x and y
    First method
    Separable, but check zero solutions first
  • Equation signal
    Right side depends on y/x
    First method
    Homogeneous substitution y=vx on x ≠ 0 interval
  • Equation signal
    y'+P(x)y=Q(x)
    First method
    Integrating factor
  • Equation signal
    Family with arbitrary constants and 'form DE'
    First method
    Advanced-explicit formation by differentiation/elimination
  • Equation signal
    Initial value supplied
    First method
    Solve general method, then determine constant, respecting domain
  • Equation signal
    Equation does not match an official first-order class
    First method
    Do not invent a method or broaden scope

Worked reasoning: separable equation with zero-solution protection

Solve dy/dx = 2xy

Known: the equation is separable, with h(y)=y.

Method choice and zero-solution protection: before dividing by y, check y=0. If y=0, then dy/dx=0 and the original equation gives 0=0, so y=0 is a valid constant solution.

For the nonzero branch: dy/y = 2x dx. Integrate: ln|y| = x²+C. Exponentiating gives y=Ce^(x²) for nonzero C during the divided derivation.

The family y=Ce^(x²) with C in the reals also includes C=0, recovering the constant solution after direct verification.

Validity check: y'=2xCe^(x²)=2xy, so every member of the stated family satisfies the original equation.

Differential-equation methods and their conditions

  • Record
    Separable structure
    Expression
    dy/dx = g(x)h(y)
    Conditions
    Functions defined on working region
    Common trap
    Dividing by h(y) immediately
    Validation
    Check h(y)=0 constant solutions first
  • Record
    Separated integration
    Expression
    ∫dy/h(y) = ∫g(x)dx + C
    Conditions
    h(y) ≠ 0 on the nonconstant branch
    Common trap
    Losing zero branch
    Validation
    Restore/check constants
  • Record
    Homogeneous substitution
    Expression
    For dy/dx=F(y/x), put y=vx, so dy/dx=v+x dv/dx
    Conditions
    Work on interval with x ≠ 0; transformed equation valid
    Common trap
    Confusing with linear homogeneous equation
    Validation
    Back-substitute
  • Record
    Linear form
    Expression
    y'+P(x)y=Q(x)
    Conditions
    P, Q valid on working interval
    Common trap
    Calling any first-order equation linear
    Validation
    Structural check
  • Record
    Integrating factor
    Expression
    IF = e^(∫P(x)dx)
    Conditions
    Standard real interval; P integrable, normally continuous
    Common trap
    Forgetting factor on Q
    Validation
    Differentiate product
  • Record
    Linear solution form
    Expression
    y·IF = ∫Q(x)IF dx + C
    Conditions
    Same interval conditions
    Common trap
    Dropping C
    Validation
    Differentiate and substitute
  • Record
    Formation from family
    Expression
    Differentiate enough times to eliminate arbitrary constants
    Conditions
    Advanced explicit; number of independent arbitrary constants guides order in standard examples
    Common trap
    Claiming every family gives exactly that order without checking dependence
    Validation
    Eliminate constants and inspect
  • Record
    Initial condition
    Expression
    Substitute (x0,y0) into the general solution
    Conditions
    Condition lies in the working interval/domain
    Common trap
    Using a point excluded by division/log
    Validation
    Check original DE and condition

Common mistakes and what they actually indicate

  • Assigning a degree when the equation is not polynomial in derivatives.

    Knowledge gap

    Why it happens

    Degree is defined only after the equation is polynomial in its derivative symbols; otherwise it is undefined.

    How it is corrected

    Check the polynomial condition on the derivative terms before stating a degree, and state 'undefined' when it fails.

  • Confusing differential-equation order with algebraic power of y.

    Knowledge gap

    Why it happens

    Order refers to the highest derivative order appearing, not any algebraic power of the function y itself.

    How it is corrected

    Identify the highest-order derivative term specifically when stating order.

  • Dividing by y or h(y) and losing constant solutions.

    Execution error

    Why it happens

    A constant solution such as y=0 can satisfy the original equation but disappears once both sides are divided by y.

    How it is corrected

    Check h(y)=0 for constant solutions before dividing, and verify whether they belong to the final family.

  • Using y=vx through x=0 without interval control.

    Execution error

    Why it happens

    The homogeneous substitution y=vx requires x ≠ 0 for dy/dx=v+x dv/dx to be valid.

    How it is corrected

    Restrict the working interval to x ≠ 0 when using the homogeneous substitution.

  • Using an integrating factor on an equation not first placed in linear form.

    Decision / selection error

    Why it happens

    The integrating factor formula applies only once the equation is written as y'+P(x)y=Q(x).

    How it is corrected

    Rearrange the equation into standard linear form before computing the integrating factor.

  • Presenting formation of differential equations as an explicit Main Unit 9 bullet.

    Needs review

    Why it happens

    Formation is explicitly listed in JEE Advanced 2026 but is not separately named in the current Main Unit 9 wording.

    How it is corrected

    Check the official-syllabus mapping table on this page before stating formation as Main-explicit scope.

FAQ

Differential Equations — questions

Straight answers about how Rank Sarthi fits into serious exam preparation.

Order is the highest derivative order appearing; degree is the highest positive-integer power of that highest-order derivative once the equation is polynomial in its derivative symbols.

Sources and provenance

Use exact source verification for worked examples; do not infer how often separable or linear equations appear. Preserve the Main/Advanced formation-scope boundary on every mention.

Contributor requirements for this page

  • Written by: unassigned. Ideal author type: Calculus/Differential Equations educator.
  • Academically reviewed by: unassigned. Reviewer specialisation: order/degree definition, formation boundary, separable zero-solution preservation, homogeneous substitution, integrating factor conditions. Minimum qualification: postgraduate degree in Mathematics, Applied Mathematics or equivalent.
  • Review scope: order/degree definition, formation boundary, separable zero-solution preservation, homogeneous substitution, integrating factor conditions, initial-condition handling and verification.