常微分方程 ODE -- Differential Equations and Solutions

Differential Equations and Solutions

Ordinary Differential Equations

An ordinary differential equation is an equation involving an unknown function of a single variable together with one or more of its derivatives. For example,
$$\frac{dy}{dt}=y-t$$
Here $y=y(t)$ is the unknown function and $t$ is the independent variable.

Order

The order of a differential equation is the order of the highest derivative that occurs in the equation.

Normal Form

General Form

$$\varphi (t,y,y^{\prime },...,y^{(n)})=0$$

But the general form is too general to deal with.

Normal Form

$$y^{(n)}=f(t,y,y^{\prime },...,y^{(n-1)})$$

Interval of Existence

The interval of existence of a solution to a differential equation is defined to be the largest interval over which the solution can be defined and remain a solution. It is important to remember that solutions to differential equations are required to be differentiable, and this implies that they are continuous.

一般初值问题需要考虑，如果解在某处不连续，要看初值是在哪段区域