Differential Equations

1) Hook — A Real-Life Story to Grab Attention

Imagine you are filling a water tank through a pipe. The rate at which water flows in depends on how much water is already in the tank — the fuller the tank, the slower the filling rate. How do we mathematically describe how the water level changes over time? This is where differential equations come into play. They help us model changing quantities and their rates, such as water levels, population growth, or even radioactive decay!

2) Core Concepts — Understanding Differential Equations

Definition: A differential equation is an equation involving an unknown function and its derivatives.

General form: If y is a function of x, then a differential equation relates y, x, and derivatives of y such as dy/dx.

Example 1: First Order Differential Equation

Consider dy/dx = 3x². This means the rate of change of y with respect to x equals 3x².

Solution: Integrate both sides:

y = ∫3x² dx = x³ + C (where C is the constant of integration)

Methods to Solve First Order Differential Equations:

• Separable Equations: Can be written as g(y) dy = f(x) dx.
• Homogeneous Equations: Functions of y/x or x/y.
• Linear Equations: Of the form dy/dx + P(x)y = Q(x).

Example 2: Solve dy/dx + y = x

This is a linear differential equation with P(x) = 1 and Q(x) = x.

Integrating factor (I.F) = e^∫P(x)dx = e^x

Multiply both sides by I.F:

e^x dy/dx + e^x y = x e^x

Left side is derivative of (y e^x):

d/dx (y e^x) = x e^x

Integrate both sides:

y e^x = ∫ x e^x dx + C

Using integration by parts, ∫ x e^x dx = (x - 1) e^x + C

Therefore, y e^x = (x - 1) e^x + C

Final solution: y = x - 1 + C e^-x

3) Key Formulas / Rules

4) Did You Know?

Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus in the late 17th century, which laid the foundation for differential equations. Today, differential equations model everything from India's population growth to the spread of diseases like COVID-19, helping scientists and policymakers make informed decisions.

5) Exam Tips — Avoid These Common Mistakes & Board Patterns

• Always write the constant of integration (C) when solving indefinite integrals.
• Check if the equation is separable or linear before choosing the method. Misidentifying can lead to wrong solutions.
• For linear equations, carefully compute the integrating factor. Missing exponentials or signs causes errors.
• Practice substitution techniques for homogeneous equations. Remember y = vx substitution.
• Previous Year Board Pattern: Usually, 2–3 questions on differential equations appear, including:
• Solving separable differential equations.
• Solving linear differential equations using integrating factor.
• Application-based problems involving rate of change.
• Time Management: Allocate 12–15 minutes per question on differential equations in the exam.