[推荐]：盐水混合模型试验报告

[推荐]：盐水混合模型试验报告 盐水混合模型试验报告 一、实验目的 1.熟悉MATLAB的运行环境. 2.学会使用MATLAB作图. 3.学会使用MATLAB编程. 二、实验内容 实验一 求解下列微分方程(组) 1. 简单微分方程 ''1); yyxyy,,,,,2,02,11，，，， '''2); (1)24,00,1210，,,,,,xyyyyy，，，，，， 2.特殊微分方程 .. 2)单摆运动方程，用图形 表 关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf 示其解。 uu，,8sin0 3.微分方程组 '1)线性微分方程组，其中矩阵的定义如下: XAX,A 010，，21，，,, ,434.,,,,,,36，，,,121，， 2)非线性微分方程组 ..2axyxyx),;,,,, .. bxxyxyyyxxy)4,4.,,,,， 实验二 盐水混合问题 一个圆柱形的容器，内装350升均匀混合的盐水溶液。如果纯水以每秒14 升的速度从容器顶部流入，同时，容器内的混合的盐水以每秒10.5升的速度从容器底部流出。开始时，容器内盐的含量为7千克。求经过时间t后容器内盐的 含量。 三、实验环境 Windows操作系统; MATLAB 7.0. 四、实验过程 1.1) dsolve('D2y=y+x-2', 'x') dsolve('D2y=y+x-2','y(0)=2,y(1)=1','x') ans = 2 - x 2) dsolve('(1+x)*D2y=(2*y-4)','x') dsolve('(1+x)*D2y=(2*y-4)','y(0)=0,y(1)-2*Dy(1)=0','x') ans = / x | / | | 1 1/2 2 + | 2 | ---------------------------------------- dy bessely(1, 2 (- 2 x - 2) ) | / 1/2 1/2 2 \ y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 1/2 1/2 1/2 bessely(1, 4 i) (x + 1) (bessely(1, 2 2 i) + 2 2 bessely(0, 4 i) i - \ / | | 1/2 | | 1/2 2 bessely(1, 4 i)) | / | bessely(1, 2 2 i) | | / \ / 0 | / | 1/2 1/2 2 | 1 | 2 + 2 bessely(1, 4 i) | ---------------------------------------- dy - | / 1/2 1/2 2 \ y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 1 / 1/2 2 | 1 2 bessely(1, 4 i) | ---------------------------------------- dy - / 1/2 1/2 2 y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 0 / 1/2 | 1 2 2 bessely(0, 4 i) bessely(1, 4 i) | ---------------------------------------- d / 1/2 1/2 2 y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 1/2 y i + 2 2 bessely(0, 4 i) bessely(1, 4 i) 1 \ \ / | | | 1 | | | ---------------------------------------- dy i | | - / 1/2 1/2 2 | | y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) / / / | | 1/2 1/2 | 2 bessely(1, 2 (- 2 x - 2) ) (x + 1) | \ / 1 | / | 1/2 1/2 2 | 1 | 2 - 2 bessely(1, 4 i) | ---------------------------------------- dy + | / 1/2 1/2 2 \ y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 0 / 1/2 | 1 bessely(1, 2 2 i) bessely(1, 4 i) | ---------------------------------------- dy / 1/2 1/2 2 y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 1 / 1/2 | + 2 2 bessely(0, 4 i) bessely(1, 4 i) | / y(1) \ \ | | 1 | | ---------------------------------------- dy i | | / 1/2 1/2 2 | | bessely(1, 2 2 (- y - 1) ) (y + 1) / / / / | | | 1/2 | 1/2 1/2 2 | bessely(1, 2 2 i) | 2 + 2 bessely(1, 4 i) | | \ \ 0 / | 1 | ---------------------------------------- dy - / 1/2 1/2 2 y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 1 / 1/2 2 | 1 2 bessely(1, 4 i) | ---------------------------------------- dy - / 1/2 1/2 2 y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 0 / 1/2 | 1 2 2 bessely(0, 4 i) bessely(1, 4 i) | ---------------------------------------- d / 1/2 1/2 2 y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) 1/2 y i + 2 2 bessely(0, 4 i) bessely(1, 4 i) 1 \ \ / | | | 1 | | | ---------------------------------------- dy i | | / 1/2 1/2 2 | | y(1) bessely(1, 2 2 (- y - 1) ) (y + 1) / / .. 2. 先做变换，将表示为关于，x,y的函数，再表示为关于t变量uu，,8sin0 的一阶微分方程 syms s t f=-8*sin(s)/t; a=16.0; b=16.0; x0=-8; y0=-8; m=40; n=40; h1=a/m; h2=b/n; hold for i=1:m s=x0+(i-1)*h1; for j=1:n t=y0+(j-1)*h2; d=eval(f); y1=t+2/3*h1*d; if abs(y1-t)>2/3*h2 x1=s+1/d*h2*2/3; plot([s,x1],[t,t+h2*2/3]) else plot([s,s+h1*2/3],[t,y1]) end end end title('d2u/dx=-8sinx');xlabel('x');ylabel('y') 则可求出其斜率场图像如下: 2du/dx=-8sinx 8 6 4 2 0 y -2 -4 -6 -8 -10-10-8-6-4-202468 x 3.1)A1: [x1,x2]=dsolve('Dx1=2*x1+x2','Dx2=-3*x1+6*x2') x1 = (C1*exp(5*t))/3 + C2*exp(3*t) x2 = C1*exp(5*t) + C2*exp(3*t) 2)A2: [x1,x2,x3]=dsolve('Dx1=x2','Dx2=4*x1+3*x2-4*x3','Dx3=x1+2*x2+x3') x1 = (3840*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) - 1920*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 1920*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) - 330*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) + 1998*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3) + 324*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) - 165*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 1998*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3) + 162*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) + 1920*3^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 1920*3^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 22*27^(1/2)*1075^(1/2)*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 165*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) + 1998*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3) - 162*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) + 165*3^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 162*3^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) + 165*3^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 162*3^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) - 11*27^(1/2)*1075^(1/2)*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 11*27^(1/2)*1075^(1/2)*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 11*3^(1/2)*27^(1/2)*1075^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 11*3^(1/2)*27^(1/2)*1075^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))))/(1458*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3)) x2 = -(6510*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) - 3255*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 3255*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) - 570*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) + 756*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3) + 162*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) - 285*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 756*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3) + 81*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) + 3255*3^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 3255*3^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 38*27^(1/2)*1075^(1/2)*C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 285*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) + 756*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3) - 81*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) + 285*3^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 81*3^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) + 285*3^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 81*3^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*((27^(1/2)*1075^(1/2))/27 + 170/27)^(4/3) - 19*27^(1/2)*1075^(1/2)*C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 19*27^(1/2)*1075^(1/2)*C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 19*3^(1/2)*27^(1/2)*1075^(1/2)*C4*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + 19*3^(1/2)*27^(1/2)*1075^(1/2)*C3*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))))/(1458*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3)) x3 = C5*exp((5*t + 12*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(9*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) + C3*cos((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) - C4*sin((5*3^(1/2)*t + 9*3^(1/2)*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3)))*exp((24*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3) - 5*t + 9*t*((27^(1/2)*1075^(1/2))/27 + 170/27)^(2/3))/(18*((27^(1/2)*1075^(1/2))/27 + 170/27)^(1/3))) 2) a)建立M文件: function xdot=fx(t,x) xdot=[-x(2)-x(1)^2;x(1)]; 然后输入: hold for k=1:7 ts=-10:0.01:10;x0=[1,0.1+0.2*k]; [t,x]=ode45('fx',ts,x0); plot(x(:,1),x(:,2)) end axis([-6 1 0 2]); 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0-6-5-4-3-2-101 b)建立M文件: function xodt=fx1(t,x) xodt=[x(1)-4*x(2)*sqrt(abs(x(1)*x(2)));-x(2)+4*x(1)*sqrt(abs(x(1)*x(2 )))]; 然后输入: hold for k=1:7 ts=-10:0.01:10;x0=[1,0.1+0.2*k]; [t,x]=ode45('fx1',ts,x0); plot(x(:,1),x(:,2)) end axis([-2 2 -2 2]); 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2-2-1.5-1-0.500.511.52 实验二 一、问题 一个圆柱形的容器，内装350升均匀混合的盐水溶液。如果纯水以每秒14升的速度从容器顶部流入，同时，容器内的混合的盐水以每秒10.5升的速度从容器底部流出。开始时，容器内盐的含量为7千克。求经过时间t后容器内盐的含量。 二、问题分析 已知:水的密度为1kg/L,盐溶解度为36g(在假设(1)中) 由题意可以知道，此题中的t时间时容器内的盐含量的问题即为关于t的微分方程的求解问题。在流入到流出的过程中，由于混合在水中的盐含量是不同的，所以溶解于水中的盐的量每一时刻都是不同的。而所求的这个瞬时量即为微分方程的解。而由题意知，可计算出7kg盐所需要的溶剂为194L水。因此，由混合液体积即可知开始时刻的7kg盐是完全溶于水中的,并且没有饱和。所以，整个过程为食盐水被再次稀释的过程，则不会出现有盐析出现象。即加入纯水盐水就被稀释，同时向外流出含有盐的混合液。因此，整体变量只与时间t有关。则可根据时间t列出微分方程求解。 三、假设 1)由 资料 新概念英语资料下载李居明饿命改运学pdf成本会计期末资料社会工作导论资料工程结算所需资料清单 知，温度对盐在水中的溶解度变化影响不大，我们可以假设盐溶解度恒为室温条件下的溶解度为36g，又由条件知该过程为稀释过程，则不存在盐析出结晶的过程，即不考虑结晶度问题。 2)任意时刻容器内混合的、流出的盐水都均匀(即任意时刻容器内及流出的盐水密度恒定包含初始状态350升混合盐水) 3)水流入及盐水流出的速度均为匀速的。 4)假设注水时间为t、t时刻时容器内含盐量为S(t)、纯水流入容器的速度为v1,混合液流出的速度为v2。 四、模型建立 计算过程如下: 1)t时刻容器内含盐量可表示为S(t) 2)纯水以v1=14L/s的速度流入容器，混合液以v2=10.5L/s的速度流出容器， m溶质则根据溶解度公式:，可推出7kg食盐是完全溶解在水中溶解度=*100gm溶剂 的 3)根据假设1)、2)，则可列出关于盐在水中的浓度的比例式，即为t时刻混合液中盐含量 *t=t,,时刻流出的混合液量时刻混合液中盐的变化量t时刻混合液量 (假设其中足够小) ,t Pt()4)根据上述比例式即可得到相应的方程为:*10.5*(),,,,tPtP(t+t)3503.5*，t dPtPt()10.5*()而使可得微分方程:。 ,,t0,,dtt3503.5*， 5)求解该微分方程当时的曲线图像 0()7,,Pt 五、模型求解 1)数值解 dsolve('Dy=-10.5*y/(350+3.5*x) ','x') ans = C2/(x + 100)^3 又由于初始x=0,时溶液中盐含量为7kg，所以可以计算出C2= 7000000 2)图形解 则带入C2= 7000000则可求出解的图像: x=linspace(0,300,10000); y= 7000000./ (x + 100).^3; plot(x,y) ,xlabel('时间'),ylabel('混合中盐含量'),title('混合液中盐含量曲线')