Chapter Hub
Differential Equations
A differential equation is an equation involving an unknown function and one or more of its derivatives. In Class 12, the focus is on identifying basic features of a differential equation, forming one from a family of curves, and solving standard first-order first-degree forms. The chapter begins with order and degree, which decide how a differential equation is classified. Degree is meaningful only when the equation can be expressed as a polynomial in derivatives after removing radicals and fractions involving derivatives. Formation of differential equations uses elimination of arbitrary constants. If a family of curves contains one arbitrary constant, differentiate once; if it contains two, differentiate twice, then eliminate the constants using the original equation and its derivatives. The solving methods in this chapter are condition-based. Variable-separable equations separate x and y terms; homogeneous equations use y = vx or x = vy; linear equations use an integrating factor. Correct identification of the form is often the key step in exams. Most errors in this chapter come from applying a method before checking its condition, missing the constant of integration, choosing the wrong integrating factor, or making algebraic mistakes during substitution and simplification.
Difficulty
Medium
Study time
70-90 min
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If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
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Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 70 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
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Key Concepts
Work through the concepts in order. Each one has an explanation, a worked example, a self-check and an exam answer.
All Concepts
All 5 concepts of the chapter, in the order the NCERT chapter teaches them.
Order and Degree of a Differential Equation
The order of a differential equation is the order of the highest derivative present. The degree is the power of the highest order derivative after the equation is made polynomial in derivatives, if such a polynomial form is possible.
Forming a Differential Equation from a Family of Curves
Forming a differential equation means eliminating arbitrary constants from a given family of curves by differentiating enough times and combining the resulting equations.
Variable-Separable Differential Equations
A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.
Homogeneous First-Order Differential Equations
A first-order differential equation dy/dx = F(x, y) is homogeneous when the right side can be expressed as a function of y/x or x/y. It is solved by substituting y = vx or x = vy to reduce it to a variable-separable equation.
Linear Differential Equations and Integrating Factor
A first-order linear differential equation in y has the form dy/dx + P(x)y = Q(x), where P and Q are functions of x. It is solved using the integrating factor e^(∫P(x) dx).
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
Common Traps
- Finding degree without first checking polynomial form in derivatives.
- Leaving arbitrary constants in the final differential equation during formation.
- Forgetting the constant of integration while solving.
- Using homogeneous substitution but omitting the product-rule term.
- Applying the linear integrating factor before converting the equation to standard form.
- Using Q instead of P while calculating the integrating factor.
- Not replacing v by y/x after solving a homogeneous equation.
How CBSE Asks This Chapter
- Order and Degree of a Differential Equation
Usually asked as a one-mark or two-mark classification question, sometimes with equations containing square roots or powers to test whether the student checks polynomial form before deciding degree.
- Forming a Differential Equation from a Family of Curves
Asked as a short-answer or long-answer item where students are given a family such as y = ax + b, y = Aex + Be−x, or circles with parameters, and must remove constants systematically.
- Variable-Separable Differential Equations
Usually asked as a direct solving problem, sometimes after rearrangement. The equation may appear as M(y)dy = N(x)dx or dy/dx = f(x)g(y).
- Homogeneous First-Order Differential Equations
Asked as a solving question where the equation is not separable at first glance. Students are expected to recognize equal-degree expressions and use y = vx or x = vy correctly.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Order and Degree of a Differential Equation
Remember
order = highest order derivative present.Trap to avoid
Students often call the highest power of any derivative the degree.
Forming a Differential Equation from a Family of Curves
Remember
Method condition: if a family contains n independent arbitrary constants, differentiate n times and eliminate those constants.Trap to avoid
A common error is differentiating but leaving the arbitrary constant in the final answer.
Variable-Separable Differential Equations
Remember
Key form: dy/dx = f(x)g(y).Trap to avoid
Students often integrate dy/dx = f(x)g(y) directly with respect to x without separating y terms correctly, which treats y like a constant.
Homogeneous First-Order Differential Equations
Remember
Condition: dy/dx = F(y/x) or dy/dx = F(x/y), or M(x, y)dx + N(x, y)dy = 0 with M and N homogeneous of the same degree.Trap to avoid
A frequent mistake is substituting y = vx but writing dy/dx = x(dv/dx), omitting the extra v term from the product rule.
Linear Differential Equations and Integrating Factor
Remember
Standard form: dy/dx + P(x)y = Q(x).Trap to avoid
Students sometimes use e^(∫Q dx) as the integrating factor.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Order and Degree of a Differential Equation | order = highest order derivative present. | Students often call the highest power of any derivative the degree. |
| Forming a Differential Equation from a Family of Curves | Method condition: if a family contains n independent arbitrary constants, differentiate n times and eliminate those constants. | A common error is differentiating but leaving the arbitrary constant in the final answer. |
| Variable-Separable Differential Equations | Key form: dy/dx = f(x)g(y). | Students often integrate dy/dx = f(x)g(y) directly with respect to x without separating y terms correctly, which treats y like a constant. |
| Homogeneous First-Order Differential Equations | Condition: dy/dx = F(y/x) or dy/dx = F(x/y), or M(x, y)dx + N(x, y)dy = 0 with M and N homogeneous of the same degree. | A frequent mistake is substituting y = vx but writing dy/dx = x(dv/dx), omitting the extra v term from the product rule. |
| Linear Differential Equations and Integrating Factor | Standard form: dy/dx + P(x)y = Q(x). | Students sometimes use e^(∫Q dx) as the integrating factor. |
Practice
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