Differential Equations - Solution by method of separation of variables

Definition of Variable Separable Form: A first-order differential equation is said to be in variable separable form if it can be expressed in the form $\frac{dy}{dx} = f(x)g(y)$. Visually, this means the rate of change can be factored into a component that depends only on the horizontal position $x$ and a component that depends only on the vertical position $y$.

The Separation Process: The fundamental step involves algebraically rearranging the equation so that all terms involving $y$ (including $dy$) are on one side of the equation and all terms involving $x$ (including $dx$) are on the other side, resulting in $\frac{1}{g(y)} dy = f(x) dx$. This conceptually decouples the variables, allowing for independent integration.

Integration and the Arbitrary Constant: Once separated, we apply the integral sign to both sides: $\int \frac{1}{g(y)} dy = \int f(x) dx + C$. The constant of integration $C$ is essential because it accounts for the fact that many functions can have the same derivative. Visually, the result describes a whole family of curves rather than a single path.

General Solution vs. Family of Curves: The resulting equation $G(y) = F(x) + C$ is called the General Solution. Geometrically, this represents an infinite set of curves in the $xy$-plane that do not intersect. Changing the value of $C$ effectively 'shifts' the curve across the plane without changing its fundamental slope characteristics at any given $(x, y)$.

Particular Solution and Initial Conditions: A Particular Solution is obtained when a specific point $(x_0, y_0)$ is provided, known as an initial condition. By substituting these coordinates into the general solution, we calculate a specific value for $C$. Graphically, this 'picks out' the single unique curve from the entire family that passes through that specific point.

Handling Logarithmic Results: In many separation problems involving $\frac{1}{y} dy$, the result is $\ln|y| = f(x) + C$. To simplify, we often express the constant as $\ln|k|$ so that the entire equation can be simplified using logarithmic properties, ultimately leading to an explicit form $y = k \cdot e^{f(x)}$.

Substitution for Non-Separable Forms: Some equations like $\frac{dy}{dx} = f(ax + by + c)$ are not initially separable. However, by using a visual transformation where we let $v = ax + by + c$, we can differentiate $v$ with respect to $x$ and substitute back into the original equation to create a new differential equation in $v$ and $x$ that is separable.