First-Order Differential Equations Calculator
Instantly solve first-order differential equations with our powerful online calculator. This tool provides comprehensive step-by-step solutions for equations that are separable and can be solved using the separation of variables technique.
Separable Equations Solver
Input your separable differential equation to find the general or particular solution, complete with detailed working and integral steps.
Separation of Variables Explained
Master the technique of separating variables with our comprehensive guide, which includes the core principles, detailed examples, and visual aids.
Separable DE Solver
Solve $dy/dx = f(x)h(y)$
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: $(2y-1)dy/dx = (3x^2+1) \implies f(x) = 3x^2+1, h(y) = 1/(2y-1)$.
Slope Field Visualizer
Visualize $dy/dx = F(x,y)$
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.
- Order of a DE: The order of the highest derivative present.
- First-order: Involves $dy/dx$ or $y'$. Ex: $\frac{dy}{dx} = 2x + y$.
- Second-order: Involves $d^2y/dx^2$ or $y''$. Ex: $y'' + 3y' + 2y = \sin(x)$.
- Solution of a DE: A function satisfying the equation.
- General Solution: Contains arbitrary constants (e.g., $C$). Represents a family of functions.
- Particular Solution: Constants are determined by initial/boundary conditions.
- Initial Condition: Specifies $y(x_0) = y_0$ for a first-order DE.
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.
