Class 12 Differential Equations Complete Guide 2027 — All Solving Methods Compared

In short: Differential Equations carries 5 marks in CBSE Class 12 Maths 2027 and tests your ability to identify and apply the correct solving method. There are three main methods: variable separable, homogeneous substitution, and linear (integrating factor). The hardest part is not solving — it is recognising which method to use. The decision flowchart below makes that automatic.

Differential Equations is Chapter 9 in CBSE Class 12 Maths and is part of the Calculus unit (35 marks total). The chapter itself carries about 5 marks — typically one 3-mark or one 5-mark question. It builds directly on Integration, so make sure your integration skills are solid before tackling this chapter.

For the complete marks breakdown, see the Class 12 Maths chapter-wise weightage.

Core Concepts

Order and Degree

Order: The highest-order derivative in the equation.

• dy/dx + y = x → Order 1
• d²y/dx² + 3dy/dx = 0 → Order 2

Degree: The power of the highest-order derivative (only when the equation is polynomial in derivatives).

• (dy/dx)³ + y = 0 → Degree 3
• d²y/dx² + sin(dy/dx) = 0 → Degree NOT defined (sin is not polynomial)

Board trap: If the highest derivative appears inside a transcendental function (sin, cos, log, exp), the degree is not defined. Students often write "degree = 1" for such cases.

Formation of Differential Equations

Given a family of curves with n arbitrary constants, differentiate n times and eliminate the constants.

Example: y = ae^x + be^(-x) has 2 constants → differentiate twice → d²y/dx² = ae^x + be^(-x) = y → d²y/dx² − y = 0.

The order of the resulting DE = number of arbitrary constants.

The Three Solving Methods

Method 1: Variable Separable

When to use: You can separate all x terms (with dx) to one side and all y terms (with dy) to the other.

Form: f(y)dy = g(x)dx

Solve: Integrate both sides: ∫f(y)dy = ∫g(x)dx + C

Example: dy/dx = x²y → dy/y = x²dx → ln|y| = x³/3 + C

Method 2: Homogeneous Differential Equations

When to use: The equation can be written as dy/dx = F(y/x), i.e., it is a function of the ratio y/x only.

Test: Replace x with tx and y with ty. If all t's cancel, it is homogeneous.

Solve: Substitute y = vx (where v = y/x), so dy/dx = v + x(dv/dx). Substitute into the equation, separate variables v and x, then integrate.

Example: (x² + y²)dx − 2xy dy = 0 → dy/dx = (x² + y²)/(2xy)

Put y = vx: v + x(dv/dx) = (1 + v²)/(2v) → separate and integrate.

Method 3: Linear Differential Equations

When to use: The equation is in the form dy/dx + Py = Q, where P and Q are functions of x only (or dx/dy + Px = Q where P, Q are functions of y only).

Solve:

1. Find the Integrating Factor: IF = e^(∫P dx)
2. Multiply the entire equation by IF
3. The solution is: y × IF = ∫(Q × IF) dx + C

Example: dy/dx + y/x = x² → P = 1/x, Q = x² IF = e^(∫1/x dx) = e^(ln x) = x Solution: yx = ∫x² · x dx + C = ∫x³ dx + C = x⁴/4 + C

Decision Flowchart — Which Method to Use

Given dy/dx = f(x,y):

Step 1: Can you separate x and y to different sides?
  YES → Variable Separable → integrate both sides
  NO → go to Step 2

Step 2: Is it of the form dy/dx = F(y/x)?
  (Replace x→tx, y→ty. Do all t's cancel?)
  YES → Homogeneous → substitute y = vx
  NO → go to Step 3

Step 3: Can you write it as dy/dx + Py = Q (P, Q functions of x)?
  YES → Linear → find IF = e^(∫P dx), apply formula
  NO → try dx/dy form (swap x and y roles) → repeat from Step 1

Memory trick: "SHL" — Separable first, Homogeneous second, Linear third. Check in this order.

Common Mistakes That Cost Marks

1. Forgetting the constant of integration. Every differential equation solution must include + C (for general solution). If an initial condition is given, use it to find C (particular solution). Missing C means incomplete answer.

2. Confusing order with degree. Order = highest derivative present. Degree = power of that highest derivative. d²y/dx² has order 2. (dy/dx)³ has degree 3. They are different concepts.

3. Not checking for homogeneity correctly. Students assume an equation is homogeneous without testing. Replace x→tx, y→ty and check if all t's cancel. If they don't, it is NOT homogeneous.

4. Wrong integrating factor. IF = e^(∫P dx) where P is the coefficient of y (not x) in the standard form dy/dx + Py = Q. Getting P wrong gives the wrong IF and the wrong solution.

5. Applying the wrong method. The most common error: trying variable separable on a linear equation, or homogeneous substitution on a separable equation. Follow the flowchart above systematically.

4-Week Practice Plan

Week 1: Order, Degree, and Formation

• Read NCERT Sections 9.1-9.3
• Practice identifying order and degree for 15 equations (include tricky ones with sin, log)
• Form DEs from given families of curves (5 problems)

Week 2: Variable Separable and Homogeneous

• Read NCERT Sections 9.4 and 9.5
• Solve 10 variable separable problems
• Solve 10 homogeneous problems (practice the y = vx substitution)
• For each problem, first decide the method, then solve

Week 3: Linear Differential Equations

• Read NCERT Section 9.6
• Solve 10 linear DE problems (focus on finding IF correctly)
• Solve 5 mixed problems where you must decide the method yourself
• Attempt CBSE Class 12 Maths important chapters questions for this topic

Week 4: Board Paper Practice

• Solve DE questions from the last 5 years of CBSE board papers
• Time yourself: 5-mark questions in 10 minutes
• Practice the decision flowchart on 10 random DEs — method identification only, no solving
• Final drill: 4 mixed DE problems (one of each type + one formation) in 30 minutes

Quick Revision Card for Exam Day

• Order = highest derivative. Degree = power of highest derivative (only if polynomial)
• Variable Separable: f(y)dy = g(x)dx → integrate both sides
• Homogeneous: dy/dx = F(y/x) → put y = vx → separate v and x
• Linear: dy/dx + Py = Q → IF = e^(∫Pdx) → y·IF = ∫Q·IF dx + C
• Decision: SHL — Separable → Homogeneous → Linear (check in order)
• Always include + C in general solutions
• Degree NOT defined if highest derivative is inside sin, cos, log, exp
• Number of constants = order of the DE

For guided practice on every DE type with method-identification hints, try Padhaao's AI tutor — it tells you which method to apply and checks each step.

Frequently Asked Questions

How many marks does Differential Equations carry in CBSE Class 12 Maths 2027?

Differential Equations carries approximately 5 marks in CBSE Class 12 Maths 2027. It is part of the Calculus unit (35 marks total). You can expect one 3-mark or one 5-mark question, usually asking you to solve a DE using a specific method or to form a DE from a given curve family.

How do I decide which method to use for solving a differential equation?

Follow the SHL order: first check if the equation is Separable (can you move all y terms to one side and x terms to the other?). If not, check if it is Homogeneous (does dy/dx = F(y/x)?). If not, check if it is Linear (can you write it as dy/dx + Py = Q?). This systematic approach works for every board exam question.

What is the difference between general and particular solutions?

A general solution contains an arbitrary constant C and represents a family of curves. A particular solution is obtained by substituting a given initial condition (like y(0) = 1) into the general solution to find the specific value of C. Board exams typically ask for the general solution unless an initial condition is given.

When is the degree of a differential equation not defined?

The degree is not defined when the highest-order derivative appears inside a transcendental function like sin, cos, tan, log, or exponential. For example, in sin(d²y/dx²) + y = 0, the degree is not defined because d²y/dx² is inside a sine function. The equation must be polynomial in its derivatives for degree to be defined.

Differential Equations is a chapter where method identification matters more than computational skill. Learn the three methods, use the SHL flowchart, and never forget the constant of integration — those 5 marks are systematic.