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8.Differential Equations – Basics
Class 12 • Mathematics • NCERT
Differential Equations – Basics
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This lesson introduces differential equations, their order and degree, and basic terminology as prescribed in the NCERT Class 12 syllabus.
Lesson Objectives
- Understand what a differential equation is.
- Learn order and degree of differential equations.
- Identify solutions of differential equations.
- Prepare for NCERT and board-level questions.
1. Meaning of Differential Equation
A differential equation is an equation involving derivatives of a dependent variable with respect to an independent variable.
Example: dy/dx = 3x²
2. Order of a Differential Equation
The order of a differential equation is the order of the highest derivative present in the equation.
For d²y/dx² + dy/dx = 0, order = 2
3. Degree of a Differential Equation
The degree of a differential equation is the power of the highest order derivative after the equation is made free from radicals and fractions of derivatives.
(dy/dx)² + y = 0 → Degree = 2
4. Solution of a Differential Equation
A solution of a differential equation is a function which satisfies the given differential equation.
For dy/dx = 2x, solution is y = x² + C
5. General and Particular Solution
A solution containing arbitrary constants is called a general solution.
A solution obtained by assigning specific values to constants is called a particular solution.
General solution: y = x² + C
Particular solution (C = 1): y = x² + 1
Particular solution (C = 1): y = x² + 1
6. Important NCERT Notes
• Order and degree are exam favourites
• Degree is defined only if equation is polynomial in derivatives
• Differential equations involve derivatives only
• Basis for solving real-life problems
• Degree is defined only if equation is polynomial in derivatives
• Differential equations involve derivatives only
• Basis for solving real-life problems
Practice Questions (NCERT)
- Define a differential equation.
- Find the order of d²y/dx² + dy/dx = 5.
- Find the degree of (dy/dx)³ = y.
- What is meant by solution of a differential equation?
- Differentiate between general and particular solution.
- Find order and degree of (d²y/dx²)² + dy/dx = 0.
- Is degree defined if derivatives are in denominator?
- Write a general solution of dy/dx = 4x.
- How many arbitrary constants appear in solution of first order DE?
- Is this chapter important for board exams?
✅ Show Answer Key
- Equation involving derivatives
- 2
- 3
- Function satisfying the DE
- General has constants, particular has fixed values
- Order = 2, Degree = 2
- No
- y = 2x² + C
- One
- Yes
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