Define differential equations, explain their importance in modeling real-world phenomena, and introduce the different types of differential equations (ordinary, partial, linear, non-linear).
Introduction to Differential Equations
Compact, structured notes for quick reference and deeper study
| Aspect | Description |
|---|---|
| Definition | An equation that relates a function (y(x)) (or several functions) to one or more of its derivatives. |
| Goal | Find the unknown function(s) that satisfy the relation. |
| Notation | Typically written as (F\big(x, y, y', y'', \dots, y^{(n)}\big)=0). |
| Order | Highest derivative appearing (e.g., first‑order, second‑order). |
| Degree | Power of the highest derivative when the equation is polynomial in derivatives (only defined for algebraic DEs). |
| Linear vs. Non‑linear | Linear: each term is either a function of (x) or a derivative multiplied by a function of (x); no products or powers of the unknown function or its derivatives. Non‑linear: any other case. |
| Category | Sub‑type | Typical Form | Example |
|---|---|---|---|
| Ordinary vs. Partial | ODE – one independent variable | (y' = f(x, y)) | (y' = xy) |
| PDE – several independent variables | (\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}) | Heat equation | |
| Order | First‑order, second‑order, … | (y^{(n)} = f(x, y, y', …, y^{(n-1)})) | (y''+3y'= \sin x) |
| Linearity | Linear | (a_n(x) y^{(n)} + … + a_0(x) y = g(x)) | (y''-4y'+4y = e^x) |
| Non‑linear | Any term like (y^2), (yy'), ((y')^2) | (y' = y^2) | |
| Homogeneous vs. Non‑homogeneous | Homogeneous (no free term) | (L[y]=0) | (y''+y=0) |
| Non‑homogeneous | (L[y]=g(x)) | (y''+y = \cos x) | |
| Exactness | Exact (total differential) | (M(x,y)dx+N(x,y)dy=0) with (\partial M/\partial y = \partial N/\partial x) | ((2xy + y^2)dx + (x^2 + 2xy)dy =0) |
| Non‑exact (requires integrating factor) | — | — |
Form: (\displaystyle \frac{dy}{dx}=g(x)h(y))
Solution steps
Example
(\displaystyle \frac{dy}{dx}=xy \Rightarrow \frac{1}{y}dy = x,dx) → (\ln|y| =