Modeling the US game (matlab self) of the differential equation

Population Model

malthus model

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Prediction Model of the US population

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Modeling and Solution

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Parameter Estimation

(1) nonlinear least squares estimation

clc, clear
a = textread('data4.txt'); %把原始数据保存在纯文本文件data4 .txt中
x=a([2:2:6],:)';%提出人口数据
x = nonzeros(x);%去掉后面的0,并变成列向量
t =[1790:10 :2000]';
t0 =t(1); x0 =x(1);
fun=@ (cs,td)cs(1)./1 +(cs(1)/x0 -1) *exp( -cs(2) * (td-t0)));%cs(1) =xm,cs(2) =r
cs = lsqcurvefit( fun, rand(2,1) ,t(2:end) ,x(2 :end) , zeros(2,1))
xhat = fun(cs,[t;2010])%预测已知年代和2010年的人口

(2) linear least squares

clc, clear
a = textread( 'data4.txt');%把原始数据保存在纯文本文件data4.txt中
x=a([2:2:6],:)'; x = nonzeros(x);
t =[1790:10 :2000]';
a=[ones(21,1)，-x(2:end)];
b=diff(x) ./x(2 :end) /10;
cs =a\b;
r =cs(1), xm=r/cs(2)

seeking differential equations matlab

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Through solutions of differential equations

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 syms y(x)
dsolve(x^2+y+(x-2*y)*diff(y)==0)

operation result:
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Initial boundary value problems

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Solving differential equations

Solving equations:

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clc, clear
syms f(x) g(x)%定义符号变量
df =diff(f);%定义f的一阶导数,用于初值或边值条件的赋值
[ f1,g1] =dsolve(diff(f,2) +3 *g= =sin(x),diff(g) +df= =cos(x))%求通解
f1 =simplify(f1), g1 =simplify(g1)%对符号解进行化简
[f2 ,g2] =dsolve(diff(f,2) +3 *g= =sin(x),diff(g) +df= =cos(x),df(2)= =0,f(3) = =3,g(5) = =1)
f2 =simplify(f2), g2 =simplify(g2)%对符号解进行化简

Solving linear differential equations

Homogeneous linear differential equations
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clc,clear
syms x(t) y(t) z(t)
X=[x;y;z];
A=[2 1 3;0 2 -1;0 0 2];
B=[1 2 1]';
[x,y,z]=dsolve(diff(X)==A*X,X(0)==B)

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Non-homogeneous linear differential equations

clc,clear
syms x(t) y(t) z(t) %定义符号变量
X=[x;y;z]; 
A=[1,0,0;2,1,-2;3,2,1];B=[0;0;exp(t) *cos(2*t)];
X0=[0;1;1];%初值条件
X =dsolve(diff(X)==A*X +B,X(0)==X0)%求符号解
X=simplify([X.x;X.y;X.z])%显示解的各个分量
pretty(x)%分数线居中的显示方式

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