📘 Differential Equations – Class 12 NCERT (With Examples)
Differential Equations are equations involving derivatives of a function. They help us understand how one quantity changes with respect to another 🔄.
🔹 1. Differential Equation
An equation containing derivatives of a dependent variable with respect to an independent variable is called a differential equation.
dy/dx = x + y is a differential equation.
🔹 2. Order of a Differential Equation
The order is the order of the highest derivative present.
d²y/dx² + 3dy/dx + y = 0
Highest derivative = d²y/dx² → Order = 2
🔹 3. Degree of a Differential Equation
The degree is the power of the highest order derivative, provided the equation is polynomial in derivatives.
(dy/dx)³ + y = 0 → Degree = 3
🔹 4. General and Particular Solutions
🧩 General Solution
A solution containing arbitrary constants is called a general solution.
dy/dx = 2x
Integrating → y = x² + C
🎯 Particular Solution
A solution obtained after assigning values to constants using given conditions.
If y = 1 when x = 0, then C = 1
Particular solution → y = x² + 1
🔹 5. Formation of Differential Equations
Differential equations are formed by eliminating arbitrary constants.
Given: y = Cx²
Differentiate: dy/dx = 2Cx
Eliminate C → x(dy/dx) = 2y
🔹 6. Solution of Differential Equations
📘 (a) Variable Separable Method
If variables can be separated on different sides, integrate both sides.
dy/dx = x/y
y dy = x dx
∫y dy = ∫x dx
y²/2 = x²/2 + C
📗 (b) Homogeneous Differential Equations
A differential equation of the form dy/dx = F(y/x) is homogeneous.
dy/dx = (x + y)/x
Put y = vx → dy/dx = v + x dv/dx
Solve to get the solution.
📙 (c) Linear Differential Equations
A first-order linear differential equation has the form:
dy/dx + Py = Q
dy/dx + y = eˣ
IF = e^(∫1 dx) = eˣ
Solution → y eˣ = ∫e²ˣ dx + C
🔹 7. Applications of Differential Equations
- 🚗 Motion with constant acceleration
- 🌱 Growth and decay of population
- 💰 Compound interest and economics
- ⚡ Electric circuits
🔹 8. Key Points to Remember
- Order ≠ Degree ❌
- Check polynomial condition for degree ✔️
- Always add constant of integration C ✍️
- Choose method wisely 🧠
✨ Conclusion
Differential Equations are powerful tools to study change in real life. Mastering this chapter helps in higher mathematics, physics, and engineering 🎓⚙️.