📐 Mathematics — Class XII · Differential Equations

Differential Equations

Equations that describe how things change — the language of nature itself

📖 Chapter 9 ⏱ ~60 min read 🏷 Differential Equations

In this chapter

Introduction
Basic Concepts
Formation of Differential Equations
Methods of Solving
Summary

9.1 Introduction

"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." — D. Hilbert

A differential equation is an equation involving derivatives of the dependent variable with respect to the independent variable. These equations arise everywhere in physics, chemistry, biology, economics — anywhere rates of change are involved.

9.2 Basic Concepts

Figure 9.1 — Order, degree, general solution, and particular solution

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Order

Highest order derivative in the equation. E.g., d²y/dx² + dy/dx = 0 has order 2.

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Degree

Power of the highest order derivative (when polynomial in derivatives). E.g., (d²y/dx²)³ has degree 3.

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General Solution

Contains arbitrary constants (as many as the order). Represents all possible solutions.

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Particular Solution

No arbitrary constants. Found by applying initial/boundary conditions to the general solution.

9.3 Formation of Differential Equations

To form a differential equation from a given family of curves:

📐 Algorithm

Step 1: Write the equation with arbitrary constants
Step 2: Differentiate as many times as the number of constants
Step 3: Eliminate the arbitrary constants from the equations
Step 4: The resulting equation is the differential equation

9.4 Methods of Solving

Methods for solving differential equations

Figure 9.2 — Three methods: variable separable, homogeneous, and linear DE

Method 1: Variable Separable

Variable Separable Form dy/dx = f(x) \cdot g(y) \to \intdy/g(y) = \intf(x) dx + C

Separate all y-terms with dy on one side, all x-terms with dx on the other, then integrate both sides.

Method 2: Homogeneous Differential Equations

Homogeneous Test F(λx, λy) = λ⁰ \cdot F(x, y) for all λ \neq 0

Substitute x = vy (or y = vx) to convert to variable separable form.

Method 3: Linear Differential Equations

Standard Form dy/dx + P(x)\cdoty = Q(x) Integrating Factor: IF = e^\intP dx Solution: y \cdot IF = \intQ \cdot IF dx + C

💡 Quick Tip

Always check in this order:
1. Is it variable separable?
2. Is it homogeneous?
3. Is it linear?

If none work, other methods (Bernoulli, exact) may be needed at higher levels.

9.5 Summary

✅ Key Takeaways

• Differential equation: involves derivatives of dependent variable

• Order: highest derivative · Degree: power of highest derivative

• General solution: has arbitrary constants (= order) · Particular: no constants

• Variable separable: separate x and y, then integrate

• Homogeneous: F(λx,λy) = λ⁰F(x,y) → substitute y = vx

• Linear DE: dy/dx + Py = Q → IF = e^∫Pdx → y·IF = ∫Q·IF dx + C

Ch 8 — Application of Integrals Ch 10 — Vector Algebra