02 - ODEs

Ordinary Differential Equation

An equation relating a variable $x$ to a function $y$ (with the finitely many derivatives of $y$ with respect to $x$)

Such that it can be written as

$$f\left(x,y,\frac{dy}{dx},\frac{d^{2}y}{d{x}^2},\cdots ,\frac{d^{k}y}{d{x}^k}\right)=0$$

Order of an ODE

If the order of an ODE is $k$ then

The ODE involves the derivatives of order $k$ or less

First Order Differential Equations

Order Differential Equations in the form of

$$\frac{dy}{dx}=f(x,y)$$

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$k$th order inhomogeneous linear ODE

They generally take the form of

$$a_k(x)\frac{d^{k}y}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}y}{d{x}^{k-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=f(x)$$

where $a_k()x)\ne 0$

$k$th order homogeneous linear ODE

Same as above but with $f(x)=0$ i.e.

$$a_k(x)\frac{d^{k}y}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}y}{d{x}^{k-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=0$$

Second Order Differential Equations