Calculus - Differential Equations (Separation of Variables)

Differential Equations Definition: A differential equation is an equation that relates a function to its derivatives. In Grade 12 Calculus, we focus on first-order equations of the form $\frac{dy}{dx} = f(x, y)$. Visually, this equation defines the slope of a tangent line at any point $(x, y)$ on a coordinate plane.

Separable Equations: A first-order differential equation is 'separable' if it can be expressed in the form $\frac{dy}{dx} = g(x)h(y)$. This means the variables $x$ and $y$ can be isolated on opposite sides of the equals sign. Graphically, the slope at any point is the product of a horizontal component and a vertical component.

The Method of Separation: To solve, rewrite the equation as $\frac{1}{h(y)} dy = g(x) dx$. We then apply the integral operator to both sides: $\int \frac{1}{h(y)} dy = \int g(x) dx$. This transforms a derivative problem into an integration problem.

The Constant of Integration ($C$): When integrating both sides, we technically get a constant on each side ($C_1$ and $C_2$), but we combine them into a single arbitrary constant $C$ on the side of the independent variable $x$. This constant is vital as it represents a family of possible solution curves rather than a single line.

General vs. Particular Solutions: The 'General Solution' contains the constant $C$ and represents an infinite family of curves. A 'Particular Solution' is found when an initial condition $(x_0, y_0)$ is provided, allowing us to solve for a specific value of $C$. Visually, the general solution is like a set of parallel-ish paths, while the particular solution is the specific path passing through a given dot.

Slope Fields (Direction Fields): A slope field is a visual representation of a differential equation. It consists of short line segments at various points $(x, y)$ where the gradient of each segment is $\frac{dy}{dx}$. If you follow the 'flow' of these segments, you can sketch the shape of the solution curves even without solving the equation analytically.

Exponential Growth and Decay: A common application is the differential equation $\frac{dy}{dx} = ky$. When separated and solved, this yields $y = Ae^{kx}$. Visually, if $k > 0$, the graph shows an upward-curving growth; if $k < 0$, it shows a downward curve that asymptotically approaches the $x$-axis.