First-Order Differential Equations

Category: Advanced Differential Calculus

1. Equations of Change

A differential equation relates an unknown function to its own derivatives. Instead of solving for a static number (like x = 5), solving a differential equation yields an entire dynamic function path framework.

2. Separable Equations Strategy

The core methodology for solving first-order variants requires grouping all instances of variable y along with its derivative token dy on one side of the equality, while locking variable x and dx on the opposite side before applying integration properties.

3. Step-by-Step Proof of Exponential Growth

Let us model and solve the equation for a system whose growth rate is directly proportional to its current size: dy/dx = ky.

Given: dy/dx = ky

Step 1: Separate variables by dividing by y and multiplying by dx:
(1/y) dy = k dx

Step 2: Take the integral of both sides of the equation:
∫ (1/y) dy = ∫ k dx

Step 3: Evaluate integration pathways:
ln|y| = kx + C

Step 4: Cancel the natural log by raising both sides to base 'e':
|y| = e^(kx + C) -> y = e^(kx) · e^C

Step 5: Define a new constant initial scaling state value A = e^C:
y = A · e^(kx)

This classic derivation mathematically accounts for compound interest curves, bacterial expansions, and radioactive half-life matrices throughout applied physics fields.

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