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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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Targeted practice
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Key Concepts
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Core Concepts
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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
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High Probability Topics
- Order and Degree of a Differential Equation
- Forming a Differential Equation from a Family of Curves
- Variable-Separable Differential Equations
- Homogeneous First-Order Differential Equations
- Linear Differential Equations and Integrating Factor
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.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, conditions, or terms
- Short answer: process, diagram, reasoning, or worked method
- Case-based: chapter scenario with linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Differential equations connect functions with their derivatives and are classified using order and degree.
- Formation of a differential equation removes arbitrary constants from a family of curves.
- Variable-separable equations are solved by separating variables and integrating both sides.
- Homogeneous equations become separable after y = vx or x = vy substitution.
- Linear equations use the integrating factor e^(integral of P dx) after conversion to standard form.
- 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 derivativ…
- 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 com…
- 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 al…
Practice
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