Differential Equations
An equation that contains variables $x$ and $y$, with at least one derivative of $y$ with respect to $x$. First-order ODEs model rates of change in physical systems.
Order and Degree
| Term | Definition |
|---|---|
| Order | The highest derivative present in the equation |
| Degree | The power of the highest derivative |
For example, $\displaystyle \left(\frac{d^2y}{dx^2}\right)^3 - \frac{dy}{dx} = 3y$ is a 2nd order, 3rd degree DE.
Classification
flowchart TD
DE["Differential Equation"] --> Order["Order"]
Order --> O1["First-Order<br/>highest derivative is dy/dx"]
Order --> O2["Second-Order<br/>highest derivative is d²y/dx²"]
Order --> O3["Nth-Order<br/>highest derivative is dⁿy/dxⁿ"]
DE --> Degree["Degree"]
Degree --> D1["First Degree<br/>highest derivative to power 1"]
Degree --> D2["Higher Degree<br/>highest derivative to power > 1"]
DE --> Linearity["Linearity"]
Linearity --> Lin["Linear<br/>y and derivatives to power 1,<br/>no products of y/derivatives"]
Linearity --> Nonlin["Non-linear<br/>products or powers ≠ 1"]
DE --> Homog["Homogeneity"]
Homog --> Hom["Homogeneous<br/>all terms contain y or its derivatives"]
Homog --> NonHom["Non-homogeneous<br/>contains standalone f(x) term"]
General and Particular Solutions
A solution to a differential equation is any function that satisfies the given equation.
- General solution: contains an arbitrary constant (e.g., $y = x^3 + C$)
- Particular solution: the constant is determined by an initial condition
First-Order ODE Solution Methods
flowchart TD
A["First-Order ODE"] --> B{"Can it be written as<br/>dy/dx = f(x) · g(y)?"}
B -->|Yes| C["Separable<br/>∫ dy/g(y) = ∫ f(x) dx"]
B -->|No| D{"Is it in the form<br/>dy/dx + P(x)y = Q(x)?"}
D -->|Yes| E{"Does it contain y^n<br/>where n ≠ 0,1?"}
E -->|Yes| F["Bernoulli Equation<br/>substitute v = y^(1-n)"]
E -->|No| G["Linear ODE<br/>Integrating Factor: μ = e^(∫P(x)dx)"]
D -->|No| H{"Can it be written as<br/>M dx + N dy = 0<br/>where ∂M/∂y = ∂N/∂x?"}
H -->|Yes| I["Exact Equation<br/>find F(x,y) = C"]
H -->|No| J["Other methods<br/>substitution, series, or numerical"]
Separable First-Order ODEs
A first-order DE is separable if it can be written in the form:
$$g(y)\frac{dy}{dx} = f(x) \quad \text{or equivalently} \quad \frac{dy}{dx} = \frac{f(x)}{g(y)}$$
Solution Method
- Separate variables: $g(y),dy = f(x),dx$
- Integrate both sides: $\displaystyle \int g(y),dy = \int f(x),dx$
- Solve for $y$ (if possible) to get the general solution
- Apply initial condition (if given) to find the particular solution
Worked Examples
Example 1 — $y\frac{dy}{dx} = 3x^2$
Separating and integrating: $\displaystyle \int y,dy = \int 3x^2,dx \Rightarrow \frac{y^2}{2} = x^3 + C$
General solution: $y^2 = 2x^3 + A$ (where $A = 2C$)
Example 2 — $x\frac{dy}{dx} = 2y$
Separating and integrating: $\displaystyle \frac{1}{2}\ln y = \ln x + C$
General solution: $y = Ax^2$ (where $A = e^{2C}$)
Example 3 (Particular Solution) — $\displaystyle \frac{dy}{dx} = \frac{2y}{x^2 - 1}$, given $y(2) = 1$
General solution: $\displaystyle y = \frac{A(x-1)}{x+1}$
Substituting $y=1, x=2$: $A = 3$
Particular solution: $\displaystyle y = \frac{3(x-1)}{x+1}$
Mixing Problems (First-Order Linear ODE)
General form: $$ \frac{dA}{dt} = \text{rate in} - \text{rate out} $$
Constant volume: $V(t) = V_0$ when inflow = outflow. Reduces to a separable/linear ODE.
Variable volume: $V(t) = V_0 + (r_{in} - r_{out})t$ when inflow ≠ outflow. Requires solving a first-order linear ODE using integrating factors.