Differential Equations
1. Basic Definitions
- Differential equation (DE): An equation that relates an unknown function with one or more of its derivatives.
- Ordinary differential equation (ODE): Contains derivatives with respect to a single independent variable.
- Partial differential equation (PDE): Contains partial derivatives with respect to multiple independent variables.
- Order: The highest derivative appearing in the equation.
- Degree: The power of the highest derivative (when the DE is expressed as a polynomial in derivatives).
- Linear DE:
- The dependent variable and its derivatives appear linearly (no products, no nonlinear functions).
- General form for ODE of order $n$: \(a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_1(x) y' + a_0(x) y = g(x).\)
- Nonlinear DE: Any DE that is not linear (e.g., $y y’ + y = 0$, $(y’’)^2 + y = 0$, $\sin(y’) = y$).
2. Classification of ODEs
| Type | Example | Characteristics |
|---|---|---|
| Separable | $\frac{dy}{dx} = \frac{x}{y}$ | $f(x)dx = g(y)dy$ |
| First-order linear | $y’ + p(x)y = q(x)$ | Integrating factor $\mu = e^{\int p dx}$ |
| Exact | $M(x,y)dx + N(x,y)dy = 0$ with $M_y = N_x$ | Potential function $\psi(x,y)=C$ |
| Homogeneous (order 0) | $y’ = F(y/x)$ | Substitution $v = y/x$ |
| Bernoulli | $y’ + p(x)y = q(x) y^n$ | Set $v = y^{1-n}$ |
| Second-order linear | $y’’ + p(x)y’ + q(x)y = r(x)$ | Superposition principle, Wronskian |
3. First‑Order ODEs – Solution Methods
3.1 Separable equations
\(\frac{dy}{dx} = g(x)h(y) \quad \Rightarrow \quad \int \frac{dy}{h(y)} = \int g(x) dx + C.\)
3.2 Linear equations
\(y' + p(x)y = q(x)\)
- Compute integrating factor $\mu(x) = e^{\int p(x) dx}$.
- Multiply both sides: $\frac{d}{dx}(\mu y) = \mu q$.
- Integrate: $y = \frac{1}{\mu}\left( \int \mu q \, dx + C \right)$.
3.3 Exact equations
$M(x,y)\,dx + N(x,y)\,dy = 0$ is exact iff $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. Find $\psi(x,y)$ such that $\psi_x = M$, $\psi_y = N$; then $\psi(x,y) = C$.
3.4 Substitution methods
- Homogeneous $y’ = F(y/x)$: let $v = y/x$.
- Bernoulli $y’ + p(x)y = q(x)y^n$: let $v = y^{1-n}$ (reduces to linear).
4. Second‑Order Linear ODEs
4.1 Homogeneous with constant coefficients
\(a y'' + b y' + c y = 0\) Characteristic equation: $a r^2 + b r + c = 0$.
| Discriminant | Roots $r_1, r_2$ | General solution |
|---|---|---|
| $>0$ | real, distinct | $y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$ |
| $=0$ | real, double $r$ | $y = (C_1 + C_2 x) e^{r x}$ |
| $<0$ | complex $\alpha \pm i\beta$ | $y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)$ |
4.2 Non‑homogeneous – method of undetermined coefficients
\(a y'' + b y' + c y = g(x)\) General solution: $y = y_h + y_p$ (homogeneous + particular).
Choose $y_p$ based on $g(x)$:
| $g(x)$ | Trial $y_p$ |
|---|---|
| $P_n(x)$ (polynomial) | $x^s Q_n(x)$ |
| $e^{\alpha x}$ | $x^s A e^{\alpha x}$ |
| $\sin(\beta x)$ or $\cos(\beta x)$ | $x^s (A\sin\beta x + B\cos\beta x)$ |
| $e^{\alpha x}\sin(\beta x)$ | $x^s e^{\alpha x}(A\sin\beta x + B\cos\beta x)$ |
The factor $x^s$ ($s=0,1,2$) is chosen so that no term in $y_p$ duplicates a solution of the homogeneous equation.
4.3 Variation of parameters (general method)
For $y’’ + p(x)y’ + q(x)y = g(x)$: \(y_p = y_1\int \frac{-y_2 g}{W} dx + y_2\int \frac{y_1 g}{W} dx,\) where $y_1, y_2$ are fundamental solutions of the homogeneous equation and $W = y_1 y_2’ - y_2 y_1’$ is the Wronskian.
5. Applications
5.1 Population growth (Malthusian)
\(\frac{dP}{dt} = kP \quad \Rightarrow \quad P(t) = P_0 e^{kt}.\)
5.2 Logistic growth
\(\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) \quad \Rightarrow \quad P(t) = \frac{K}{1 + \left(\frac{K}{P_0} - 1\right)e^{-rt}}.\)
5.3 Newton’s law of cooling
\(\frac{dT}{dt} = -k(T - T_{\text{amb}}) \quad \Rightarrow \quad T(t) = T_{\text{amb}} + (T_0 - T_{\text{amb}})e^{-kt}.\)
5.4 RLC circuit (second order)
\(L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t) \quad \text{or} \quad L\frac{di}{dt} + Ri + \frac{1}{C}\int i\,dt = E(t).\)
5.5 Harmonic oscillators
- Undamped: $m x’’ + kx = 0$ → simple harmonic motion.
- Damped: $m x’’ + c x’ + kx = 0$ (over‑, critical‑, under‑damped).
- Forced: $m x’’ + c x’ + kx = F_0 \cos(\omega t)$ → resonance when $\omega \approx \sqrt{k/m}$.
6. Laplace Transforms (for IVPs)
Definition: \(\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) dt.\)
Useful transforms
| $f(t)$ | $F(s)$ |
|---|---|
| $1$ | $\frac{1}{s}$ |
| $t^n$ | $\frac{n!}{s^{n+1}}$ |
| $e^{at}$ | $\frac{1}{s-a}$ |
| $\sin(\omega t)$ | $\frac{\omega}{s^2+\omega^2}$ |
| $\cos(\omega t)$ | $\frac{s}{s^2+\omega^2}$ |
| $t^n e^{at}$ | $\frac{n!}{(s-a)^{n+1}}$ |
| $f’(t)$ | $sF(s) - f(0)$ |
| $f’‘(t)$ | $s^2F(s) - s f(0) - f’(0)$ |
| $e^{at}f(t)$ | $F(s-a)$ (first shifting theorem) |
Solving IVPs using Laplace
- Apply $\mathcal{L}$ to both sides of the ODE.
- Use initial conditions and table of transforms.
- Solve for $Y(s) = \mathcal{L}{y(t)}$.
- Compute inverse Laplace (partial fractions + tables).
7. Systems of ODEs (optional)
A system of first‑order linear ODEs can be written as \(\mathbf{x}' = A \mathbf{x}, \quad \mathbf{x}(t) \in \mathbb{R}^n,\) with solution $\mathbf{x}(t) = e^{At} \mathbf{x}(0)$, where $e^{At}$ is the matrix exponential.
For non‑homogeneous systems $\mathbf{x}’ = A\mathbf{x} + \mathbf{f}(t)$, use variation of parameters or Laplace transforms.
8. Quick Checklist for Solving ODEs
- Identify order and linearity.
- For first order: try separation → linear → exact → substitution.
- For second order constant coefficients: write characteristic equation.
- For non‑homogeneous: choose undetermined coefficients (if $g(x)$ is simple) or variation of parameters.
- For IVPs with discontinuous inputs or initial conditions at 0, use Laplace transform.
- Check your solution by plugging back into the ODE.
Happy solving!