微分方程建模

微分方程建模

原文：http://blog.youkuaiyun.com/qq_34861102/article/details/76974294

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• 对dsolve所得的解析解画图：

  y = dsolve();
  ezplot(y);

• 阶层

  gamma(n+1) = n!

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• 微分方程

  

  这里只是求曲线，可以用到的函数dsolve

  

  代码：

  y = dsolve('D2y = sqrt(1+(Dy)^2)/5/(1-x)','y(0) = 0,Dy(0) = 0','x');
  
  dyy = @(x,yy)[yy(2);sqrt(1+yy(2)^2)/5/(1-x)];
  yy0 = [ 0 0 ]';
  [x yy] = ode45(dyy,[0 ,0.99999],yy0);
  plot(x,yy(:,1));
  yys = yy(end,1)

  

  • 求数值解的时候2阶微分方程是不存在的，需要做变量变换：

    

    clc,clear
    y = dsolve('x^2*D2y+x*Dy+(x^2-1/4)*y','y(pi/2)=2,Dy(pi/2)=-2/pi','x');
    ezplot(y);
    hold on;
    %定义函数
    dy = @(x,y)[y(2);(1/4/x^2-1)*y(1)-y(2)/x];
    [x,y] = ode45(dy,[pi/2,8],[2,-2/pi]);
    plot(x,y(:,1),'*');

  • 列表内容

    

    clc,clear
    yy = @(x)1 - 1/gamma(3)*x.^2 + 2/gamma(5)*x.^4 - 9/gamma(7)*x.^6 + 55/gamma(9)*x.^8;
    x1 = 0:0.1:2;
    y1 = yy(x1);
    plot(x1,y1,'P -');
    hold on;
    dy = @(x,y)[y(2);-y(1)*cos(x)];
    [x2,y2] = ode45(dy,[0,2],[1,0]);
    plot(x2,y2(:,1),'* -r')

    

    dxy = @(t,xy)[-2*xy(1)/sqrt(xy(1)^2+xy(2)^2);1-2*xy(2)/sqrt(xy(1)^2+xy(2)^2)];
    [t,xy] = ode45(dxy,[0,66.65],[100,0]);
    plot(xy(:,1),xy(:,2));