1、1. 对于二阶全微分方程 a ，不同的 a,b,c 取值会求出不同的解2+=()析解，解析解又是由齐次解和特解组成。其中，齐次解由特征方程决定，而特解的决定因素则比较复杂。2. 对于二阶全微分方程的分析，我们大致分为三种情况：b2-4ac0(两个不同的实根 )b2-4ac=0（两个相同的重根）b2-4ac0 的情况STEP1：求解析解s1=dsolve(D2y+3*Dy+2*y=0,y(0)=2,Dy(0)=0,t);s2=dsolve(D2y+3*Dy+2*y=sin(t),y(0)=2,Dy(0)=0,t);s3=dsolve(D2y+3*Dy+2*y=sin(2*t),y(0)=2,Dy
2、(0)=0,t);s4=dsolve(D2y+3*Dy+2*y=sin(5*t),y(0)=2,Dy(0)=0,t);s5=dsolve(D2y+3*Dy+2*y=sin(13*t),y(0)=2,Dy(0)=0,t);s6=dsolve(D2y+3*Dy+2*y=sin(25*t),y(0)=2,Dy(0)=0,t);STEP2：绘制图形（1 ）求 w=1 情况下的通解和齐次解t=1:0.1:10;s1=4*exp(-t)-2*exp(-2*t) %general solutions2=-3/10*cos(t)+1/10*sin(t)-11/5*exp(-2*t)+9/2*exp(-t) %
3、special solutionsubplot(2,1,1);plot(t,s2);xlabel(t)ylabel(y(t)title(general solution )subplot(2,1,2);plot(t,s1);xlabel(t)ylabel(y(t)title(special solution)Figure1-1(2)求通解随 w 变化的规律.w在（ 0,1）之间的全微分方程通解clcclear alls1=dsolve(D2y+3*Dy+2*y=0,y(0)=2,Dy(0)=0,t);l2=dsolve(D2y+3*Dy+2*y=sin(0.05*t),y(0)=2,Dy(0)
4、=0,t);l3=dsolve(D2y+3*Dy+2*y=sin(0.15*t),y(0)=2,Dy(0)=0,t);l4=dsolve(D2y+3*Dy+2*y=sin(0.25*t),y(0)=2,Dy(0)=0,t);l5=dsolve(D2y+3*Dy+2*y=sin(0.5*t),y(0)=2,Dy(0)=0,t);l6=dsolve(D2y+3*Dy+2*y=sin(0.75*t),y(0)=2,Dy(0)=0,t);t=1:0.1:10;s1_n = eval(s1);l2_n = eval(l2);l3_n = eval(l3);l4_n = eval(l4);l5_n = e
5、val(l5);l6_n = eval(l6);hold onplot(t,s1_n);plot(t,l2_n,m*);plot(t,l3_n,rx);plot(t,l4_n,g);plot(t,l5_n,bp);plot(t,l6_n,ko);hold off.w 在（1,+ ）之间的全微分方程通解t=1:0.1:10;s1=-2*exp(-2*t)+4*exp(-t);s2=-3/10*cos(t)+1/10*sin(t)-11/5*exp(-2*t)+9/2*exp(-t);s3 =-3/20*cos(2*t)-1/20*sin(2*t)-9/4*exp(-2*t)+22/5*exp(-
6、t);s4 =-15/754*cos(5*t)-23/754*sin(5*t)-63/29*exp(-2*t)+109/26*exp(-t);s5=693/170*exp(-t)-39/29410*cos(13*t)-359/173*exp(-2*t)-167/29410*sin(13*t);s6=-1283/629*exp(-2*t)-75/393754*cos(25*t)+2529/626*exp(-t)-623/393754*sin(25*t);hold onplot(t,s1);plot(t,s2,m*);plot(t,s3,rx);plot(t,s4,g);plot(t,s5,bp)
7、;plot(t,s6,ko);hold off结论：在 b2-4ac0 的情况下，特解的形式是 C1*sint+C2*cost,齐次解的形式是C1*EXP(R1*t)+C2*EXP(R2*t).若 w 为正值且随 w 的增大，通解的形式趋近于齐次解。5.b2-4ac=0 的情况STEP1：求解析解S1=dsolve(D2y+y=0,y(0)=2,Dy(0)=1,t)S2=dsolve(D2y+y=sin(t),y(0)=2,Dy(0)=1,t)s3=dsolve(D2y+y=sin(2*t),y(0)=2,Dy(0)=1,t)s4=dsolve(D2y+y=sin(6*t),y(0)=2,Dy
8、(0)=1,t)s5=dsolve(D2y+y=sin(10*t),y(0)=2,Dy(0)=1,t)s6=dsolve(D2y+y=sin(100*t),y(0)=2,Dy(0)=1,t)l0=dsolve(D2y+y=sin(0.05*t),y(0)=2,Dy(0)=1,t)l1=dsolve(D2y+y=sin(0.15*t),y(0)=2,Dy(0)=1,t)l2=dsolve(D2y+y=sin(0.25*t),y(0)=2,Dy(0)=1,t)l3=dsolve(D2y+y=sin(0.5*t),y(0)=2,Dy(0)=1,t)l4=dsolve(D2y+y=sin(0.75*t
9、),y(0)=2,Dy(0)=1,t)STEP2：绘制图形（1） 求 w=1 情况下的通解和齐次解t=1:0.1:10;s1=2*exp(-2*t)+5*exp(-2*t).*t;s2=54/25*exp(-2*t)+26/5*exp(-2*t).*t-4/25*cos(t)+3/25*sin(t);subplot(2,1,1);plot(t,s1);xlabel(t)ylabel(y(t)title(homogenious solution)subplot(2,1,2);plot(t,s2);xlabel(t)ylabel(y(t)title(general solution )（2） (2
10、)求通解随 w 变化的规律.w在（ 0,1）之间的全微分方程通解t=1:0.1:10;s1=2*exp(-2*t)+5*exp(-2*t).*t;l2=5158402/2563201*exp(-2*t)+8025/1601*exp(-2*t).*t-32000/2563201*cos(1/20*t)+639600/2563201*sin(1/20*t);l3=5273762/2588881*exp(-2*t)+8105/1609*exp(-2*t).*t-96000/2588881*cos(3/20*t)+636400/2588881*sin(3/20*t);l4=8706/4225*exp(
11、-2*t)+329/65*exp(-2*t).*t-256/4225*cos(1/4*t)+1008/4225*sin(1/4*t);l5=610/289*exp(-2*t)+87/17*exp(-2*t).*t-32/289*cos(1/2*t)+60/289*sin(1/2*t);l6=11426/5329*exp(-2*t)+377/73*exp(-2*t).*t-768/5329*cos(3/4*t)+880/5329*sin(3/4*t);hold onplot(t,s1);plot(t,l2,m*);plot(t,l3,rx);plot(t,l4,g);plot(t,l5,bp);
12、plot(t,l6,ko);hold off.w在（ 1,+ ）之间的全微分方程通解t=1:0.1:10;s1=2*exp(-2*t)+5*exp(-2*t).*t;s2=54/25*exp(-2*t)+26/5*exp(-2*t).*t-4/25*cos(t)+3/25*sin(t);s3=3522/1681*exp(-2*t)+215/41*exp(-2*t).*t-160/1681*cos(5/2*t)-36/1681*sin(5/2*t);s4=350/169*exp(-2*t)+68/13*exp(-2*t).*t-12/169*cos(3*t)-5/169*sin(3*t);s5=
13、1702/841*exp(-2*t)+150/29*exp(-2*t).*t-20/841*cos(5*t)-21/841*sin(5*t);s6=104942/52441*exp(-2*t)+1160/229*exp(-2*t).*t-60/52441*cos(15*t)-221/52441*sin(15*t);hold onplot(t,s1);plot(t,s2,m*);plot(t,s3,rx);plot(t,s4,g);plot(t,s5,bp);plot(t,s6,ko);hold off结论： W 属于（0,1）时，随 w 的增大在齐次解的旁边波动；w 属于（1 ，+ ），随 w
14、 的增大逐渐趋近于齐次解。4. b2-4ac0s2=dsolve(D2y+Dy+y=sin(t),y(0)=2,Dy(0)=1,t)s3=dsolve(D2y+Dy+y=sin(2*t),y(0)=2,Dy(0)=1,t)s4=dsolve(D2y+Dy+y=sin(2.5*t),y(0)=2,Dy(0)=1,t)s5=dsolve(D2y+Dy+y=sin(3*t),y(0)=2,Dy(0)=1,t)s6=dsolve(D2y+Dy+y=sin(3.5*t),y(0)=2,Dy(0)=1,t)s7=dsolve(D2y+Dy+y=sin(5*t),y(0)=2,Dy(0)=1,t).w在（
15、0,1）之间的全微分方程通解.w在（ 1,+ ）之间的全微分方程通解2.b.w在（ 0,1）之间的全微分方程通解.w在（ 1,+ ）之间的全微分方程通解2.b=0STEP1：求解析解S1=dsolve(D2y+y=0,y(0)=2,Dy(0)=1,t)S2=dsolve(D2y+y=sin(t),y(0)=2,Dy(0)=1,t)s3=dsolve(D2y+y=sin(2*t),y(0)=2,Dy(0)=1,t)s4=dsolve(D2y+y=sin(6*t),y(0)=2,Dy(0)=1,t)s5=dsolve(D2y+y=sin(10*t),y(0)=2,Dy(0)=1,t)s6=dsol
16、ve(D2y+y=sin(100*t),y(0)=2,Dy(0)=1,t)l0=dsolve(D2y+y=sin(0.05*t),y(0)=2,Dy(0)=1,t)l1=dsolve(D2y+y=sin(0.15*t),y(0)=2,Dy(0)=1,t)l2=dsolve(D2y+y=sin(0.25*t),y(0)=2,Dy(0)=1,t)l3=dsolve(D2y+y=sin(0.5*t),y(0)=2,Dy(0)=1,t)l4=dsolve(D2y+y=sin(0.75*t),y(0)=2,Dy(0)=1,t)STEP2：绘制图形（3） 求 w=1 情况下的通解和齐次解t=1:0.1:1
17、0;s1=sin(t)+2*cos(t);s2 =3/2*sin(t)+2*cos(t)-1/2.*cos(t).*t;subplot(2,1,1);plot(t,s1);xlabel(t)ylabel(y(t)title(homogenious solution)subplot(2,1,2);plot(t,s2);xlabel(t)ylabel(y(t)title(general solution )(2)求通解随 w 变化的规律.w在（ 0,1）之间的全微分方程通解t=1:0.1:10;l0=379/399*sin(t)+2*cos(t)+400/399*sin(1/20*t);l1=33
18、1/391*sin(t)+2*cos(t)+400/391*sin(3/20*t);l2=11/15*sin(t)+2*cos(t)+16/15*sin(1/4*t);l3=1/3*sin(t)+2*cos(t)+4/3*sin(1/2*t);l4 =-5/7*sin(t)+2*cos(t)+16/7*sin(3/4*t);s1=sin(t)+2*cos(t);hold onplot(t,s1);plot(t,l0,m*);plot(t,l1,rx);plot(t,l2,g);plot(t,l3,bp);plot(t,l4,ko);hold off.w在（ 1,+ ）之间的全微分方程通解t=1
19、:0.1:10;l2=11/15*sin(t)+2*cos(t)+16/15*sin(1/4*t);l3=1/3*sin(t)+2*cos(t)+4/3*sin(1/2*t);s1=sin(t)+2*cos(t);%s2 =3/2*sin(t)+2*cos(t)-1/2*cos(t)*t;s3 =5/3*sin(t)+2*cos(t)-1/3*sin(2*t);s4 =41/35*sin(t)+2*cos(t)-1/35*sin(6*t);hold onplot(t,s1);plot(t,s3,rx);plot(t,s4,g);plot(t,l2,bp);plot(t,l3,ko);hold off结论： w=1 是特殊情况，s2 =3/2*sin(t)+2*cos(t)-1/2*cos(t)*t（见figure）； W 属于（0,1）时，随 w 的增大在齐次解的旁边波动；w 属于（1，+ ），随 w 的增大逐渐趋近于齐次解。
