常微分方程 ODE -- Differential Equations and Solutions

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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

$\phi (t,y,y', ..., y^{(n)})=0$

But the general form is too general to deal with.

Normal Form

$y^{(n)}=f(t,y,y',...,y^{(n-1)})$

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