通过MATLAB求二阶全微分方程解析解

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(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)%generalsolutions2=-3/10*cos(t)+1/10*sin(t)-11/5*exp(-2*t)+9/2*exp(-t)%specialsolutionsubplot(2,1,1);plot(t,s2);xlabel('t')ylabel('y(t)')title('generalsolution')subplot(2,1,2);plot(t,s1);xlabel('t')ylabel('y(t)')title('specialsolution')Figure1-1.w(2)求通解随w变化的规律

clc在（0,1）之间的全微分方程通解clears1=dsolve(alll2=dsolve('D2y+3*Dy+2*y=0'l3=dsolve('D2y+3*Dy+2*y=sin(0.05*t)','y(0)=2,Dy(0)=0',,'t');l4=dsolve('D2y+3*Dy+2*y=sin(0.15*t)','y(0)=2,Dy(0)=0''y(0)=2,Dy(0)=0',,'t''t'););l5=dsolve('D2y+3*Dy+2*y=sin(0.25*t)''D2y+3*Dy+2*y=sin(0.5*t)',,'y(0)=2,Dy(0)=0''y(0)=2,Dy(0)=0','y(0)=2,Dy(0)=0',,'t''t');,'t');t=1:0.1:10;l6=dsolve('D2y+3*Dy+2*y=sin(0.75*t)');s1_nl2_n=eval(s1);l3_n=eval(l2);l4_n=eval(l3);l5_n=eval(l4);l6_n=hold=eval(l5);eval(l6);plot(t,s1_n);onplot(t,l2_n,plot(t,l3_n,'m*');plot(t,l4_n,'rx'plot(t,l5_n,'g^'););plot(t,l6_n,'bp''ko'););holdoff.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);s3s4=-3/20*cos(2*t)-1/20*sin(2*t)-9/4*exp(-2*t)+22/5*exp(-t);=-15/754*cos(5*t)-23/754*sin(5*t)-63/29*exp(-2*t)+109/26*exp(-t);(13*t);s5=693/170*exp(-t)-39/29410*cos(13*t)-359/173*exp(-2*t)-167/29410*sins6=-1283/629*exp(-2*t)-75/393754*cos(25*t)+2529/626*exp(-t)-623/393754*sin(25*t);holdplot(t,s1);onplot(t,s2,plot(t,s3,'m*');plot(t,s4,'rx');plot(t,s5,'g^'plot(t,s6,'bp'););holdoff'ko');结论：在b^2-4ac>0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.b^2-4ac=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(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)','y(0)=2,Dy(0)=1','t')STEP2：绘制图形t=1:0.1:10;（1）求w=1情况下的通解和齐次解

s1=2*exp(-2*t)+5*exp(-2*t).*t;subplot(2,1,1);plot(t,s1);s2=54/25*exp(-2*t)+26/5*exp(-2*t).*t-4/25*cos(t)+3/25*sin(t);xlabel('t')ylabel('y(t)')title('homogenioussolution')subplot(2,1,2);plot(t,s2);xlabel('t')ylabel('y(t)')title('generalsolution')（2）(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;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);(3/20*t)+6300/2588881*sin(3/20*t);225*sin(1/4*t);1/2*t);l3=5273762/2588881*exp(-2*t)+8105/1609*exp(-2*t).*t-96000/2588881*cosl4=8706/4225*exp(-2*t)+329/65*exp(-2*t).*t-256/4225*cos(1/4*t)+1008/4l5=610/2*exp(-2*t)+87/17*exp(-2*t).*t-32/2*cos(1/2*t)+60/2*sin(l6=11426/5329*exp(-2*t)+377/73*exp(-2*t).*t-768/5329*cos(3/4*t)+880/5329*sin(3/4*t);holdonplot(t,s1);plot(t,l2,'m*');plot(t,l3,'rx');plot(t,l4,'g^');plot(t,l5,plot(t,l6,'bp'holdoff'ko'););.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);1*sin(5/2*t);s3=3522/1681*exp(-2*t)+215/41*exp(-2*t).*t-160/1681*cos(5/2*t)-36/168;s4=350/169*exp(-2*t)+68/13*exp(-2*t).*t-12/169*cos(3*t)-5/169*sin(3*t)s5=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);holdplot(t,s1);onplot(t,s2,plot(t,s3,'m*'plot(t,s4,'rx'););plot(t,s5,'g^'plot(t,s6,'bp');holdoff'ko'););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的增大逐渐趋近于齐次解。4.b^2-4ac<0的情况

1.[b>0]

s2=dsolve(s3=dsolve('D2y+Dy+y=sin(t)'s4=dsolve('D2y+Dy+y=sin(2*t)','y(0)=2,Dy(0)=1''D2y+Dy+y=sin(2.5*t)','y(0)=2,Dy(0)=1','t','y(0)=2,Dy(0)=1','t'),'t')s5=dsolve('D2y+Dy+y=sin(3*t)','y(0)=2,Dy(0)=1','t'))s7=dsolve(s6=dsolve('D2y+Dy+y=sin(5*t)''D2y+Dy+y=sin(3.5*t)','y(0)=2,Dy(0)=1','y(0)=2,Dy(0)=1','t','t')).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<0].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=0]STEP1：求解析解S1=dsolve(S2=dsolve('D2y+y=0','y(0)=2,Dy(0)=1','t')s3=dsolve('D2y+y=sin(t)'s4=dsolve('D2y+y=sin(2*t)','y(0)=2,Dy(0)=1','t')s5=dsolve('D2y+y=sin(6*t)',,'y(0)=2,Dy(0)=1','t')s6=dsolve('D2y+y=sin(10*t)''y(0)=2,Dy(0)=1'l0=dsolve('D2y+y=sin(100*t)','y(0)=2,Dy(0)=1','t'),'y(0)=2,Dy(0)=1','t',)l1=dsolve('D2y+y=sin(0.05*t)''D2y+y=sin(0.15*t)',,'y(0)=2,Dy(0)=1''t''y(0)=2,Dy(0)=1',,'t')'t'))l2=dsolve(l3=dsolve('D2y+y=sin(0.25*t)','y(0)=2,Dy(0)=1','t')l4=dsolve('D2y+y=sin(0.5*t)''D2y+y=sin(0.75*t)','y(0)=2,Dy(0)=1','y(0)=2,Dy(0)=1','t','t'))STEP2：绘制图形（3）求w=1情况下的通解和齐次解

t=1:0.1:10;s1=sin(t)+2*cos(t);subplot(2,1,1);s2=3/2*sin(t)+2*cos(t)-1/2.*cos(t).*t;plot(t,s1);xlabel(ylabel('t')title(subplot(2,1,2);'homogenious'y(t)')solution')plot(t,s2);xlabel(ylabel('t'title('general'y(t)'))solution')(2).w求通解随w变化的规律

t=1:0.1:10;在（0,1）之间的全微分方程通解l0=379/399*sin(t)+2*cos(t)+400/399*sin(1/20*t);l1=331/391*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);l4s1=sin(t)+2*cos(t);=-5/7*sin(t)+2*cos(t)+16/7*sin(3/4*t);holdplot(t,s1);onplot(t,l0,plot(t,l1,'m*'plot(t,l2,'rx');plot(t,l3,'g^');plot(t,l4,'bp');holdoff'ko'););.w在（1,+）之间的全微分方程通解t=1:0.1:10;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);%s2s3=3/2*sin(t)+2*cos(t)-1/2*cos(t)*t;s4=5/3*sin(t)+2*cos(t)-1/3*sin(2*t);hold=41/35*sin(t)+2*cos(t)-1/35*sin(6*t);plot(t,s1);onplot(t,s3,plot(t,s4,'rx''g^'););plot(t,l2,plot(t,l3,'bp');holdoff'ko');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。