MathIsimple

Differential Equations Fundamentals

Master first-order differential equations including separable equations and linear ODEs. Learn solution methods, integrating factors, and real-world applications in growth models and circuits.

12th Grade
Calculus
~90 min
🎮 Interactive Activity: Separable ODE Solver

Solve this separable differential equation!

Solve:

dy/dx = 2xy

Separate variables and integrate

🎮 Interactive Activity: Integrating Factor Calculator

Find the integrating factor for this linear ODE!

For the equation:

y' + 2y = Q(x)

Find μ(x):

1. Introduction to Differential Equations

What Are Differential Equations?

A differential equation is an equation that relates a function to its derivatives. First-order differential equations involve only the first derivative dy/dx.

Example 1: Basic Form

General form: F(x, y, y') = 0

First-order: Involves only y and y' (first derivative)

Example: dy/dx = 2x + y

Solution: A function y(x) that satisfies the equation

Example 2: Initial Value Problem

Problem: dy/dx = 2x, y(0) = 1

Solution: y = x² + C, then y(0) = 1 gives C = 1

Final answer: y = x² + 1

Key point: Initial conditions determine the constant

2. Separable Differential Equations
3. Linear First-Order Differential Equations
4. Integrating Factor Method
5. Applications: Exponential Growth and Decay
6. Logistic Growth Model
7. Initial Value Problems
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
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Correct
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Score
1
What is a separable differential equation?
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2
What is the integrating factor for y' + 2y = x?
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3
What is the general solution to dy/dx = ky?
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4
What is an initial value problem?
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5
What is the logistic growth model?
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6
What method is used to solve linear first-order ODEs?
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7
What is the solution to dy/dx = 2xy with y(0) = 1?
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8
What does 'ODE' stand for?
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9
What is the order of a differential equation?
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10
Which real-world phenomenon is modeled by dy/dx = -ky?
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