数学建模作业

第一题：建立一个符号表达式y=sin((a+b)*x)，以变量x从pi/2到pi进行积分程序：clearsymsabxf=sym('sin((a+b)*x)')int(f,x,pi/2,pi)结果：f=sin((a+b)*x)ans=-(cos(pi*a)*cos(pi*b)-sin(pi*a)*sin(pi*b)-cos(1/2*pi*a)*cos(1/2*pi*b)+sin(1/2*pi*a)*sin(1/2*pi*b))/(a+b)第二题：求符号矩阵F对变量x的一阶微分、对变量a的二阶微分F=[cos(a*x)sin((a+b)*x)][-sin(b*x)400];程序：clearsymsabxf=sym('[cos(a*x),sin((a+b)*x);-sin(b*x),400]')diff(f,x,1)iff(f,a,2)结果：f=[cos(a*x),sin((a+b)*x)][-sin(b*x),400]ans=[-sin(x*a)*a,cos((a+b)*x)*(a+b)][-cos(x*b)*b,0]ans=[-cos(x*a)*x^2,-sin((a+b)*x)*x^2][0,0]第三题:求解下列微分方程a.y’=(x+y)(x-y)b.xy’=ytg(y/x)，y(10)=1(a)程序：dsolve('Dy=(x+y)*(x-y)','x')结果:ans=x*(C1*besseli(-3/4,1/2*x^2)-besselk(3/4,1/2*x^2))/(C1*besseli(1/4,1/2*x^2)+besselk(1/4,1/2*x^2))(b)定义M文件:functionDy=wang(x,y)Dy=y*tan(y/x)/x;end程序：[x,y]=ode45('wang',[1,100],10);plot(y,x);运行结果：第四题：求微分方程组的解。42fggff程序：[f,g]=dsolve('Df=f+2*g','Dg=f+4')结果：f=-exp(-t)*C2+2*exp(2*t)*C1-4g=exp(-t)*C2+exp(2*t)*C1+2第五题：求微分方程组，当初始条件为f(0)=2，g(2)=5时的解,并画出解f(t),g(t)的图像。程序：[f,g]=dsolve('Df=f+g','Dg=-f+g','f(0)=2','g(2)=5')结果：f=exp(t)*((2*cosh(2)*sin(2)+2*sinh(2)*sin(2)+5)/cos(2)/(cosh(2)+sinh(2))*sin(t)+2*cos(t))g=exp(t)*((2*cosh(2)*sin(2)+2*sinh(2)*sin(2)+5)/cos(2)/(cosh(2)+sinh(2))*cos(t)-2*sin(t))ezplot(f)ezplot(g)gftfddgftgdd