如何解高阶时变微分方程？关于ode45的用法~。

ode45 help 文档关于解时变微分方程有如下描述

Example 3

This example solves an ordinary differential equation withtime-dependent terms.

Consider the following ODE, with time-dependent parameters definedonly through the set of data points given in two vectors:

y'(t) + f(t)y(t) = g(t)

Theinitial condition is y(0) = 0, where the function f(t) isdefined through the n-by-1 vectors tf and f,and the function g(t) is defined through the m-by-1 vectors tg and g.

First, define the time-dependent parameters f(t) and g(t) asthe following:

ft = linspace(0,5,25); % Generate t for f

f = ft.^2 - ft - 3; % Generate f(t)

gt = linspace(1,6,25); % Generate t for g

g = 3*sin(gt-0.25); % Generate g(t)

Write a function to interpolate the data sets specifiedabove to obtain the value of the time-dependent terms at the specifiedtime:

function dydt = myode(t,y,ft,f,gt,g)

f = interp1(ft,f,t); % Interpolate the data set (ft,f) at time t

g = interp1(gt,g,t); % Interpolate the data set (gt,g) at time t

dydt = -f.*y + g; % Evalute ODE at time t

Call the derivative function myode.m withinthe MATLAB ode45 function specifying timeas the first input argument :

Tspan = [1 5]; % Solve from t=1 to t=5

IC = 1; % y(t=0) = 1

[T Y] = ode45(@(t,y) myode(t,y,ft,f,gt,g),Tspan,IC); % Solve ODE

Plot the solution y(t) as a functionof time:

plot(T, Y);

title('Plot of y as a function of time');

xlabel('Time'); ylabel('Y(t)');

但是关于高阶方程，比如2阶，必须要引入状态量进行降阶，变成1阶才能用ode45

比如

y''+c(t)y'+my=f(t)

令 x1=y

x2=y'

x3=y''

原微分方程化为

x1'=x2

x2'=-c(t)x2-kx1+f(t)方程组

第二个方程化为1阶段，但是引入了一个x1啊。在使用ode45所述的例子的时候，如何处理？

Answer 1

[sacobeh]

你这里没有必要引入 x3，只要 x1、x2就够了。也就是原1元2阶微分方程转化为了2元1阶微分方程组：

x1'=x2

x2'=-c(t)x2-kx1+f(t)

对这个微分方程组直接用ode求解，但你需要的只是x1的解（因为 x1 就是 y）。 返回值的第一列对应 x1，也就是你要求的 y，第二列是 x2'，可以忽略