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,
d y d t = y − t \frac{dy}{dt} = y-t dtdy=y−t
Here y = y ( t ) y = y(t) y=y(t) is the unknown function and t t 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
ϕ ( t , y , y ′ , . . . , y ( n ) ) = 0 \phi (t,y,y', ..., y^{(n)})=0 ϕ(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 ) ) y^{(n)}=f(t,y,y',...,y^{(n-1)}) 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