重庆大学数学实验实验三

重庆大学学生实验报告实验课程名称数学实验开课实验室学院年级专业班学生姓名学号开课时间至学年第学期总成绩教师签名数学与统计学院制开课学院、实验室：实验时间：2013年4月24日课程名称数学实验实验项目名称第三次实验实验项目类型验证演示综合设计其他指导教师成绩实验目的学会了使用dsolve命令求解微分方程的解析解；会使用ode23和ode45命令求解微分方程及微分方程组的的数值解；会通过画相图来分析一些简单的问题。基础实验一、实验内容见第三次实验内容。二、实验过程（一般应包括实验原理或问题分析，算法设计、程序、计算、图表等，实验结果及分析）1.第一题程序：y1=dsolve('Dy=y+2*x','y(0)=1','x')subplot(221),ezplot(y1,[0,1])y2=dsolve('D2y=-y+cos(x)','y(0)=1','Dy(0)=0','x')subplot(222),ezplot(y2)[x3,y3]=dsolve('Dx=a*x+b*y','Dy=a*x-b*y','x(0)=2','y(0)=0')a=1;b=1;x=sym('x3');y=sym('y3');subplot(223),ezplot(x,y)第一题运行结果：y1=-2*x-2+3*exp(x)y2=cos(x)+1/2*sin(x)*xx3=(a+b+(a^2+6*b*a+b^2)^(1/2))/(a^2+6*b*a+b^2)^(1/2)*exp(1/2*(a-b+(a^2+6*b*a+b^2)^(1/2))*t)-(a+b-(a^2+6*b*a+b^2)^(1/2))/(a^2+6*b*a+b^2)^(1/2)*exp(1/2*(a-b-(a^2+6*b*a+b^2)^(1/2))*t)y3=-1/2*((a+b+(a^2+6*b*a+b^2)^(1/2))/(a^2+6*b*a+b^2)^(1/2)*exp(1/2*(a-b+(a^2+6*b*a+b^2)^(1/2))*t)*a+(a+b+(a^2+6*b*a+b^2)^(1/2))/(a^2+6*b*a+b^2)^(1/2)*exp(1/2*(a-b+(a^2+6*b*a+b^2)^(1/2))*t)*b-(a+b+(a^2+6*b*a+b^2)^(1/2))*exp(1/2*(a-b+(a^2+6*b*a+b^2)^(1/2))*t)-(a+b-(a^2+6*b*a+b^2)^(1/2))/(a^2+6*b*a+b^2)^(1/2)*exp(1/2*(a-b-(a^2+6*b*a+b^2)^(1/2))*t)*a-(a+b-(a^2+6*b*a+b^2)^(1/2))/(a^2+6*b*a+b^2)^(1/2)*exp(1/2*(a-b-(a^2+6*b*a+b^2)^(1/2))*t)*b-(a+b-(a^2+6*b*a+b^2)^(1/2))*exp(1/2*(a-b-(a^2+6*b*a+b^2)^(1/2))*t))/b2.第二题程序：（1）先编写M文件：functionweif=erone(t,x)weif=sin(t)*x+exp(sin(t));在外面窗口中运行下面命令:ode45('erone',[0,5],2.35)第（1）题运行结果：（2）编写M文件：functionweif=ertwo(t,x)weif=[x(2);-(x(1)-t)*x(2)./(t+1)-x(1)./(t+1)+(t*cos(t))./(t+1)];在外面的窗口中运行下面的命令：[t,x]=ode23('ertwo',[0,1],[2,-1])plot(t,x)第（2）题运行结果：t=00.10000.20000.30000.40000.50000.60000.70000.80000.90001.0000x=2.0000-1.00001.9000-1.00001.8000-1.00021.7000-1.00081.5998-1.00221.4995-1.00511.3987-1.00991.2974-1.01741.1951-1.02831.0916-1.04340.9863-1.0636第三题程序：首先编写rossler.M文件：functionf=rossler(t,x)globalaf=[-x(2)-x(3);x(1)+a*x(2);2+x(3)*(x(1)-4)];然后再外面窗口编写如下命令：globalafora=0.1:0.1:0.6[t,x]=ode45('rossler',[0,200],[0,0,0]);subplot(121);plot(t,x(:,1),'r',t,x(:,2),'g',t,x(:,3),'b');title('x(红),y(绿),z(蓝)随t的变化情况');xlabel('t');subplot(122);plot3(x(:,1),x(:,2),x(:,3));title('相图');xlabel('x');ylabel('y');zlabel('z');pauseend第三题运行结果：a=0.1时a=0.2时a=0.3时a=0.4时a=0.5时a=0.6时运行出错，输不出结果。4.第四题程序：编写si.m的M文件：functionf=si(t,x)f=[x(2)+cos(t);sin(2*t)];在外面的窗口中编写下面命令：[t,x]=ode45('si',[0,50],[0.5,-0.5])plot(x(:,2),x(:,1))title('x1-x2Ïàͼ');xlabel('x2');ylabel('x1');第四题程序运行结果：t=00.05020.10050.15070.20100.38930.57760.76590.95421.19991.44561.69131.93702.16932.40162.63392.86623.10903.35173.59453.83724.04844.25974.47094.68224.88265.08315.28355.48405.68045.87696.07336.26986.46406.65836.85257.04687.26767.48847.70927.93008.17168.41318.65478.89639.14639.39649.64659.896510.116310.336110.555910.775611.017411.259211.501011.742811.938312.133712.329212.524612.719912.915113.110313.305613.523513.741413.959314.177214.420914.664614.908315.152015.403415.654915.906316.157816.378816.599816.820817.041717.237217.432617.628017.823418.018918.214318.409718.605118.819619.034119.248619.463219.668619.874120.079620.285120.542620.800221.057821.315321.557821.800322.042822.285322.512822.740322.967823.195323.386723.578023.769423.960724.152124.343424.534824.726124.957525.188825.420225.651525.851526.051426.251326.451226.697426.943627.189827.436027.670227.904428.138628.372928.614328.855729.097129.338529.571229.803930.036630.269230.458230.647130.836031.024931.257331.489631.721931.954332.155132.355932.556732.757533.005933.254333.502733.751233.986334.221534.456734.691934.930035.168235.406335.644435.833836.023136.212436.401836.591136.780436.969837.159137.384837.610537.836238.062038.256938.451838.646838.841739.076039.310339.544639.778940.009740.240540.471340.702140.956941.211841.466641.721441.932642.143842.355042.566242.754842.943443.131943.320543.525843.731043.936244.141544.333544.525544.717544.909645.157945.406245.654545.902846.135746.368746.601646.834547.076047.317547.559047.800548.012048.223548.435048.646648.849449.052149.254949.457749.593349.728949.864450.0000x=0.5000-0.50000.5251-0.49750.5504-0.48990.5759-0.47750.6018-0.46020.7039-0.35570.8174-0.20190.9434-0.01981.08000.16561.26370.36911.43040.48461.55210.48551.60100.37181.55890.18171.4231-0.04551.1986-0.26300.9028-0.42600.5486-0.49860.1897-0.4565-0.1336-0.3091-0.3868-0.0893-0.53000.1207-0.59570.3088-0.58700.4425-0.51450.4991-0.40180.4711-0.26320.3688-0.11390.20800.03320.01380.1666-0.17890.2864-0.34390.3937-0.45640.4934-0.49980.5914-0.46740.6961-0.36580.8123-0.20960.9419-0.02181.10290.19431.26720.37231.41810.47891.53520.49421.59850.40161.57240.21851.4459-0.01451.2221-0.24570.9065-0.42470.5426-0.49940.1736-0.4518-0.1568-0.2935-0.3831-0.0927-0.53220.1247-0.59730.3183-0.58230.4524-0.48840.4991-0.33970.4321-0.16350.26610.01590.03820.1502-0.15500.2712-0.32430.3795-0.44460.4793-0.49820.5775-0.47640.6814-0.38320.7965-0.23230.9250-0.04611.08260.16891.24520.35151.39730.46841.51950.49841.59480.42081.58160.24671.46700.01521.2523-0.22130.9426-0.41030.5798-0.49730.2072-0.4612-0.1303-0.3109-0.3647-0.1129-0.52210.1058-0.59480.3040-0.58580.4449-0.51940.4982-0.41210.4766-0.27860.3831-0.13330.23160.01090.04490.1457-0.14790.2672-0.31840.3758-0.44140.4853-0.49890.5938-0.46630.7102-0.34920.8408-0.16840.98170.03411.13310.23031.28540.38801.42450.48181.55380.48431.60110.36261.54270.14691.3702-0.10861.1105-0.32410.7832-0.46420.4235-0.49710.0718-0.4158-0.2138-0.2508-0.4296-0.0360-0.55980.1861-0.60020.3716-0.56970.4697-0.49120.4998-0.37720.4577-0.24190.3493-0.09900.19010.04080.00390.1704-0.18270.2869-0.34350.4119-0.46960.5289-0.49690.6486-0.41980.7813-0.25410.9115-0.06601.05450.13181.20370.30881.34830.43771.49810.49961.58750.44361.59100.28231.49310.05301.3023-0.17791.0308-0.36940.7034-0.48120.3536-0.49020.0102-0.3878-0.2774-0.1983-0.48100.0365-0.58550.2646-0.59150.4271-0.51870.4985-0.38760.4639-0.22270.3307-0.08150.16870.0557-0.01650.1824-0.19940.2963-0.35470.4209-0.47500.5381-0.49460.6591-0.40940.7938-0.23710.9260-0.04571.07070.15221.22070.32591.36440.44841.51100.49921.59300.43021.58520.25731.47290.02111.2689-0.20830.9865-0.39140.6523-0.48940.3010-0.4820-0.0322-0.3666-0.3066-0.1710-0.49670.0625-0.58970.2836-0.59420.4159-0.54520.4891-0.45320.4930-0.33130.4271-0.19360.3005-0.05220.13170.0835-0.05560.2079-0.23560.3391-0.40460.4571-0.49230.5708-0.48130.6905-0.37400.8063-0.21940.9358-0.03261.07670.15921.22240.32761.38870.46301.52090.49851.59430.42671.58770.26261.48900.04481.2993-0.18111.0312-0.36900.7083-0.48080.3277-0.4857-0.0296-0.3685-0.3209-0.1577-0.51500.0948-0.59010.2880-0.59020.4302-0.52500.4967-0.41020.4763-0.28140.3865-0.14170.2428-0.00200.06500.1296-0.12230.2585-0.30590.3734-0.43820.4788-0.49770.5819-0.47490.6842-0.38140.7974-0.23280.9238-0.05031.06170.14001.24760.35231.41870.47881.54660.48971.60230.38261.56650.19641.4366-0.03031.2169-0.25040.9240-0.41850.5733-0.49740.2152-0.4630-0.1102-0.3228-0.3684-0.1080-0.51860.1025-0.59130.2941-0.58870.4338-0.52100.4977-0.40990.4760-0.27110.3776-0.12030.21790.02890.02240.1228-0.11230.2104-0.23860.2912-0.34750.3661-0.4311第五题程序：先编写名为logistic.m的M文件：functionf=logistic(t,x)globalaba=0.5;b=2;r=[11];n=[100100];m=[b*x(1)a*x(2)];f=[r(1)*x(1)*(1-(x(1)+m(2))/n(1));r(2)*x(2)*(1-(x(2)+m(1))/n(2))];在外面的窗口中运行下面的程序命令：[t,x]=ode45('logistic',[0,20],[10,10])subplot(121);plot(t,x(:,1),'r',t,x(:,2),'g')title('种群随时间的变化情况');xlabel('时间t');ylabel('种群数量');gtext('x(1)');gtext('x(2)');subplot(122);plot(x(:,1),x(:,2))title('相图');xlabel('x(1)');ylabel('x(2)');第五题运行结果：（1）a=1.5b=2时，t=00.05910.11820.17730.23640.52210.80781.09361.37931.75562.13192.50822.88463.20913.53363.85814.18264.45064.71874.98685.25495.47255.69015.90786.12546.34306.56076.77836.99597.21367.43147.64917.86688.07298.27898.48508.69118.89249.09369.29499.49629.69519.894110.093010.291910.489710.687510.885311.083011.280211.477411.674611.871812.068712.265612.462512.659312.856113.052813.249613.446313.643013.839614.036314.233014.429614.626214.822915.019515.233115.446815.660415.874116.122916.371716.620516.869317.161517.453817.746018.038318.390618.742819.095119.447419.585619.723719.861920.0000x=10.000010.000010.513010.418011.047510.844011.603911.277312.182811.717115.306213.903118.986716.073023.218818.056627.958319.674134.836820.957842.193521.106849.734720.133457.154418.241863.246016.148868.906513.863774.029311.571078.57559.421981.89397.828284.81296.421487.34825.208989.52654.185591.05513.482692.38782.883693.54292.378294.53891.954495.39391.600496.12471.306996.74691.065197.27490.866597.72160.703298.09850.569898.41540.461498.68130.373498.89320.305199.07210.249199.22280.203499.34960.166099.45390.136099.54190.111499.61600.091299.67820.074799.73010.061399.77370.050399.81030.041299.84110.033899.86690.027899.88850.022899.90660.018799.92180.015499.93460.012699.94530.010499.95420.008599.96170.007099.96800.005799.97320.004799.97760.003999.98130.003299.98440.002699.98700.002199.98910.001899.99090.001499.99240.001299.99370.001099.99470.000899.99560.000799.99630.000599.99690.000499.99750.000499.99790.000399.99830.000299.99860.000299.99880.000299.99900.000199.99920.000199.99940.000199.99950.000199.99960.000099.99970.000099.99980.000099.99980.000099.99990.000099.99990.000099.99990.0000100.00000.0000100.00000.0000100.00000.0000100.00000.0000100.00000.0000100.00000.0000分析图可知，种群2竞争能力较弱，最终灭亡，种群1的竞争能力较强，最终达到环境所容许的最大值。（2）a=1.5b=0.7时将logistic.m文件中的ab的值改为a=1.5b=0.7，然后保存，运行程序，可得结果：t=00.06050.12110.18160.24210.54100.83981.13871.43751.82292.20832.59372.97913.27993.58073.88164.18244.48324.78405.08485.38565.73496.08436.43366.78307.22687.67068.11448.55829.05829.558210.058210.558210.988411.418611.848812.279012.687013.094913.502913.910814.310714.710615.110415.510315.906616.302816.699117.095317.489917.884518.279118.673719.005319.336919.668420.0000x=10.000010.000010.460510.512610.934011.045711.420011.599811.917912.174914.529015.327417.2990