1st Order Differential Equation Solver: Separation of Variables

$dy/dx = F(x,y)$

Solve $dy/dx = f(x)h(y)$

$f(x)$ part (e.g., x, 1, sin(x)):

$h(y)$ part (e.g., y, 1, 1/y, exp(y)):

For $dy/dx = f(x)$, set $h(y)=1$. For $dy/dx = h(y)$, set $f(x)=1$.

For $(A(y))\frac{dy}{dx} = B(x)$, input $f(x) = B(x)$ and $h(y) = 1/A(y)$.

Ex: $(2y-1)dy/dx = (3x^2+1) \implies f(x) = 3x^2+1, h(y) = 1/(2y-1)$.

Initial Condition $x_0$ (optional):

Initial Condition $y_0$ (optional):

Slope Field Visualizer

Visualize $dy/dx = F(x,y)$

$F(x,y)$ (e.g., x-y, sin(x), y/x):

x min:

x max:

y min:

y max:

Differential Equation Definitions

A differential equation (DE) is an equation that relates one or more unknown functions and their derivatives. DEs are fundamental to describing various phenomena in science, engineering, economics, and biology.

Core Concepts for Solving 1st Order Differential Equations

When utilizing a 1st Order Differential Equation Solver (Separation of Variables), understanding key terminology is paramount. This section clarifies the fundamental concepts that define and influence the solution process for differential equations.

Understanding the Order of a Differential Equation

The 'order' of a differential equation is a critical classification that dictates its complexity and, crucially, the methods applicable for its solution. It is determined by the highest derivative present within the equation.

First-Order (1st Order) Differential Equations

These are the primary focus of a 1st Order Differential Equation Solver. They involve only the first derivative of an unknown function, typically expressed as $dy/dx$ or $y'$. 1st order differential equations are foundational for modeling dynamic systems where a quantity's rate of change is directly dependent on its current state.

Example: $\frac{dy}{dx} = 2x + y$.
Relevance: Techniques like the separation of variables are specifically designed and highly effective for solving certain types of these 1st order differential equations.

Second-Order Differential Equations

These equations include the second derivative ($d^2y/dx^2$ or $y''$) as their highest derivative. They are used to model more complex systems involving acceleration, oscillations, or vibrations.

Example: $y'' + 3y' + 2y = \sin(x)$.
Note: While important, these are beyond the scope of a basic 1st Order Differential Equation Solver that specializes in separation of variables.

The Nature of a Differential Equation Solution

A 'solution' to a differential equation is a function that, when substituted back into the original equation, satisfies it. The ultimate goal of any 1st Order Differential Equation Solver is to accurately find these functions.

General Solution

The general solution to a differential equation is a family of functions that all satisfy the equation. For a 1st order differential equation, this solution typically contains one arbitrary constant (e.g., $C$). When you use a 1st Order Differential Equation Solver and apply the separation of variables method, the initial outcome after integration is always the general solution.

Particular Solution

A particular solution is a unique function derived from the general solution. It is obtained by determining the value(s) of the arbitrary constant(s) using specific conditions provided with the differential equation, such as initial or boundary conditions. This allows for a precise model of a specific scenario.

The Role of Initial Conditions in 1st Order DEs

For a 1st order differential equation, an 'initial condition' is a specific constraint that helps pinpoint a unique particular solution from the infinite family of functions represented by the general solution.

Definition: An initial condition typically specifies the value of the dependent variable $y$ at a given value of the independent variable $x$, often written as $y(x_0) = y_0$.
Application: When provided to a 1st Order Differential Equation Solver, this condition is used to solve for the arbitrary constant $C$ found during the separation of variables process, thus yielding a unique and specific solution to the problem.

Solving Separable DEs: $dy/dx = f(x)h(y)$

A first-order DE is separable if it can be written with all $y$ terms on one side and all $x$ terms on the other.

Given: $$ \frac{dy}{dx} = f(x)h(y) $$

1. Separate variables: Assuming $h(y) \neq 0$: $$ \frac{1}{h(y)} \, dy = f(x) \, dx $$

2. Integrate both sides: $$ \int \frac{1}{h(y)} \, dy = \int f(x) \, dx $$

3. General Solution (often implicit): If $H(y)$ is an antiderivative of $1/h(y)$ and $F(x)$ is an antiderivative of $f(x)$: $$ H(y) = F(x) + C $$ If $h(y_0)=0$, then $y=y_0$ is a constant (equilibrium) solution.

Special Cases:

If $dy/dx = f(x)$ (i.e., $h(y)=1$): $y = \int f(x)dx = F(x)+C$.
If $dy/dx = h(y)$ (i.e., $f(x)=1$): $\int \frac{1}{h(y)}dy = \int dx \implies H(y) = x+C$.

Beyond Separable Equations

This solver focuses on separable first-order DEs. Other types include:

Linear Equations: $\frac{dy}{dx} + P(x)y = Q(x)$ (solved using an integrating factor).
Exact Equations: $M(x,y)dx + N(x,y)dy = 0$ where $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
Homogeneous Equations: $\frac{dy}{dx} = F(y/x)$ (substitute $v=y/x$).
Bernoulli Equations: $\frac{dy}{dx} + P(x)y = Q(x)y^n$.

These require different techniques not covered by this specific solver.