通过MATLAB求二阶全微分方程解析解

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：

b^2-4ac>0(两个不同的实根)

b^2-4ac=0（两个相同的重根）

b^2-4ac<0（两个不同的复数根）

对三种情况进行MATLAB编程，分析齐次解和特解后，再改变W的值，观察解析解的变化

3.b^2-4ac>0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(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 solution

s2=-3/10*cos(t)+1/10*sin(t)-11/5*exp(-2*t)+9/2*exp(-t)%special solution

subplot(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

clc

clear all

s1=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)=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=eval(l5);

l6_n=eval(l6);

hold on

plot(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(-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)-623/39375 4*sin(25*t);

hold on

plot(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：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

（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)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)+6300/2588881*sin(3/20*t);

l4=8706/4225*exp(-2*t)+329/65*exp(-2*t).*t-256/4225*cos(1/4*t)+1008/4 225*sin(1/4*t);

l5=610/2*exp(-2*t)+87/17*exp(-2*t).*t-32/2*cos(1/2*t)+60/2*sin( 1/2*t);

l6=11426/5329*exp(-2*t)+377/73*exp(-2*t).*t-768/5329*cos(3/4*t)+880/5 329*sin(3/4*t);

hold on

plot(t,s1);

plot(t,l2,'m*');

plot(t,l3,'rx');

plot(t,l4,'g^');plot(t,l5,'bp');

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)-36/168 1*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=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)-22 1/52441*sin(15*t);

hold on

plot(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的增大逐渐趋近于齐次解。

4.b^2-4ac<0的情况

1.[b>0]

s2=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（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('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

（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);

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=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);l4=-5/7*sin(t)+2*cos(t)+16/7*sin(3/4*t); s1=sin(t)+2*cos(t);

hold on

plot(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=1:0.1:10;

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 on

plot(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。