Chapter 9 — Differential Equations — covers order, degree, formation, and solving differential equations (variable separable, homogeneous, linear). Carries 6-8 marks.
Key Concepts
Order and Degree
Order: Highest order derivative present
Degree: Power of the highest order derivative (after removing radicals/fractions)
Degree is defined only when DE is polynomial in derivatives
Degree: Power of the highest order derivative (after removing radicals/fractions)
Degree is defined only when DE is polynomial in derivatives
Methods of Solving
1. Variable Separable
dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx
2. Homogeneous Equations
dy/dx = F(y/x) — substitute y = vx, dy/dx = v + x(dv/dx)
Separate variables in v and x, then integrate
Separate variables in v and x, then integrate
3. Linear Differential Equation
Form: dy/dx + Py = Q (where P, Q are functions of x)
Integrating Factor (IF): e^(∫P dx)
Solution: y × IF = ∫(Q × IF) dx + C
Also: dx/dy + Px = Q (P, Q functions of y): IF = e^(∫P dy)
Integrating Factor (IF): e^(∫P dx)
Solution: y × IF = ∫(Q × IF) dx + C
Also: dx/dy + Px = Q (P, Q functions of y): IF = e^(∫P dy)
Solved Examples
Example 1: Variable Separable
Q: Solve dy/dx = (1+y²)/(1+x²)
Solution: dy/(1+y²) = dx/(1+x²)
∫dy/(1+y²) = ∫dx/(1+x²)
tan⁻¹y = tan⁻¹x + C
Example 2: Linear DE
Q: Solve dy/dx + y/x = x²
Solution: P = 1/x, Q = x². IF = e^(∫1/x dx) = e^(ln x) = x
y·x = ∫x²·x dx = ∫x³ dx = x⁴/4 + C
y = x³/4 + C/x
Quick Revision Points
- Order = highest derivative; Degree = power of highest order derivative
- Variable separable: f(x)dx = g(y)dy
- Homogeneous: put y = vx; converts to separable
- Linear: dy/dx + Py = Q; IF = e^(∫P dx); Solution: y·IF = ∫Q·IF dx + C
- Number of arbitrary constants = order of DE
Chapter Navigation
Previous: Application of Integrals Class 12 Notes
Next: Vector Algebra Class 12 Notes
Related Chapters in Class 12 Maths
- Integrals Class 12 Notes
- Continuity and Differentiability Class 12 Notes
- Application of Derivatives Class 12 Notes
Practice What You Learned
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