01 Intro
DE¶
Differential equation is an equation that relates one or more unknown functions and their derivatives.
Order¶
The order of a differential equation is defined to be that of the highest order derivative it contains.
Degree¶
The degree of a differential equation is defined as the power to which the highest order derivative is raised.
1st Order DE¶
\[ \frac{dy}{dx} = f(x, y) \]
Aim is to find the value of \(y\) in terms in \(x\). We do this by integrating (anti-derivative).
Separable Variables¶
We can directly solve this by separating the variables
\[ \begin{aligned} f(x, y) &= g(x) \cdot h(y) \\ \frac{dy}{dx} &= g(x) \cdot h(y) \\ \int \frac{dy}{h(y)} &= \int g(x) dx \end{aligned} \]
Homogeneous Equation¶
Special type of Homogeneous Expression
\[ M dx + N dy = 0 \]
If both \(M(x, y)\) and \(N(x,y)\) are homogeneous of the same degree.
\[ \begin{aligned} \frac{dy}{dx} &= \frac{- M(x, y)}{N(x, y)} \\ \text{Let } v &= \frac{y}{x} \implies y = vx \\ \frac{dy}{dx} &= v + x \frac{dv}{dx} \end{aligned} \]
Homogeneous Expression¶
\[ \begin{aligned} f(tx, ty) = t^n \cdot f(x, y) \end{aligned} \]
| Example | Degree |
|---|---|
| \(\sin(\frac{x}{y})\) | 0 |
| \(\sqrt{x^2 + y^2}\) | 1 |
| \(x^2 + y^2\) | 2 |
Integration Rules¶
Transposed Differential¶
Be able to identify transposition to simplify
\[ \begin{aligned} d(\log x) &= \left( \frac{1}{x} \right) dx \\ \text{because } \frac{d(\log x)}{dx} &= \frac{1}{x} \\ \end{aligned} \]