Differential Equations
Board: CBSE | Class: Class 12
Comprehensive study notes for Differential Equations by Ajay Yadav.
Key Concepts
Order and Degree
Order: highest derivative in equation. Degree: highest power of highest derivative (after removing radicals). Example: (y”)³+(y’)²=0: Order=2, Degree=3.
Formation of DE
Eliminate arbitrary constants from family of curves. Number of constants = order of DE.
Variable Separable
If dy/dx = f(x)g(y): separate variables → dy/g(y) = f(x)dx. Integrate both sides.
Homogeneous DE
dy/dx = f(y/x). Substitute y=vx → dy/dx = v + x(dv/dx). Then separate variables.
Linear DE
dy/dx + P(x)y = Q(x). Integrating factor (IF) = e∫Pdx. Solution: y(IF) = ∫Q(IF)dx + C.
Important Formulas
| Variable separable | dy/g(y) = f(x)dx |
| Homogeneous | Substitute y=vx |
| Linear DE | IF = e∫Pdx, y(IF)=∫Q(IF)dx+C |
Solved Examples
Example 1: Find order/degree of (y”)²+y’=0.
Solution: Order=2, Degree=2.
Example 2: Solve dy/dx = x/y.
Solution: y dy = x dx ⇒ y²/2 = x²/2 + C ⇒ y²-x²=C.
Example 3: Solve dy/dx + y = ex.
Solution: Linear, P=1, Q=ex. IF=e∫dx=ex. y(ex)=∫e²xdx = e²x/2 + C. y=ex/2+Ce⁻x.
Practice Questions
- Find order/degree: (y”)³+x(y’)⁴+y=0.
- Solve: dy/dx = (1+y²)/(1+x²).
- Solve: dy/dx = (x+y)/x (homogeneous).
- Solve: dy/dx + 2xy = x.
- Form DE of family y=Ae²x+Be⁻²x.
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