《高等应用数学问题的MATLAB求解》——第7章习题代码

（1）原例题以及练习

>> syms t;u=sin(t);uu=diff(u)+3u;
>> y=dsolve(['D5y+5D4y+12D3y+16D2y+12Dy+4y=',char(uu)],'y(0)=0','Dy(0)=0','D2y(0)=0','D3y(0)=0','D4y(0)=0')
>> %Support of character vectors and strings will be removed in a future release. Use sym objects to define differential equations instead.
结果

\[{\mathrm{e}}^{-t}-\frac{\cos\left(t\right)}{5}-\frac{2\,\sin\left(t\right)}{5}-\frac{4\,{\mathrm{e}}^{-t}\,\cos\left(t\right)}{5}+\frac{11\,{\mathrm{e}}^{-t}\,\sin\left(t\right)}{10}-\frac{t\,{\mathrm{e}}^{-t}\,\cos\left(t\right)}{2} \]

>> syms t x(t) y(t); uu=diff(sin(t))+3sin(t);
>> y1=diff(y);y2=diff(y,2);y3=diff(y,3);y4=diff(y,4);y5=diff(y,5);
>> z1=dsolve(y5+5y4+12y3+16y2+12y1+4yuu,y(0)0,y1(0)0,y2(0)0,y3(0)0,y4(0)0);
>> z1=simplify(z1)
结果

\[{\mathrm{e}}^{-t}-\frac{\cos\left(t\right)}{5}-\frac{2\,\sin\left(t\right)}{5}-\frac{4\,{\mathrm{e}}^{-t}\,\cos\left(t\right)}{5}+\frac{11\,{\mathrm{e}}^{-t}\,\sin\left(t\right)}{10}-\frac{t\,{\mathrm{e}}^{-t}\,\cos\left(t\right)}{2} \]

（2）

>> syms t y(t); u=exp(-2t)( sin(2t+pi/3)+cos(3t) );
>> y1=diff(y);y2=diff(y1);y3=diff(y2);y4=diff(y3);y5=diff(y4);
>> z=dsolve(y5+13y4+64y3+152y2+176y1+80*yu,y(0)1,y(1)3,y(pi)2,y1(0)1,y1(1)2);
>> z=simplify(z);latex(z)

（3）

①

>> syms t x(t) y(t);
>> [x1 y1]=dsolve(diff(x,t,2)-2diff(y,t,2)+diff(y,t)+x-3y0,4diff(y,t,2)-2diff(x,t,2)-diff(x,t)-2*x5*y)
②

>> syms t x(t) y(t);
>> [x1 y1]=dsolve(2diff(x,t,2)+2diff(x,t,1)-x+3diff(y,t,2)+diff(y,t,1)+y==0,diff(x,t,2)+4diff(x,t,1)-x+diff(y,t,2)+2*diff(y,t)-y==0)

（4）

>> syms t x(t) y(t);
>> [x1 y1]=dsolve(diff(x,t,2)+5diff(x,t)+4x+3y==exp(-t)sin(4t),2diff(y,t)+y+4diff(x,t)+6xexp(-t)cos(4t),x(0)1,x(pi)2,y(0)0)

（5）

1

>> syms t x(t) n;
>> x1=dsolve((1-t^2)diff(x,t,2)-2tdiff(x)+n(n+1)*x==0,t)

2

>> syms t x(t) n;
>> x1=dsolve(t^{2*diff(x,t,2)+t*diff(x,t)+(t}2-n^2)*x==0)

（6）

>> syms x y(x);
>> y1=simplify(dsolve(diff(y,x,2)-(2-1/x)diff(y,x)+(1-1/x)yx^2exp(-5x)))
>> y2=simplify(dsolve(diff(y,x,2)-(2-1/x)diff(y,x)+(1-1/x)yx^2exp(-5x),y(1)pi,y(pi)1))

（8）

①

>> syms t x(t); x1=dsolve(diff(x,t,2)+2tdiff(x,t)+t^2*x==t+1)

②

>> syms x y(x); y1=dsolve(diff(y,x)+2xy==x*exp(-x^2))

③

>> syms t y(t); y1=dsolve(diff(y,t,3)+3diff(y,t,2)+3diff(y,t)+y==exp(-t)*sin(t))

（9）

①

>> syms x y(x); y1=dsolve(diff(y,x)==y^4*cos(x)+y*tan(x))

②

>> syms x y(x);y1=dsolve(x*y^{2*diff(y,x)==x}2+y^2)
③

>> syms x y(x);y1=dsolve(xdiff(y,x)+2y+x^5*y3*exp(x)==0)

（10）

>> syms t x(t) y(t);
>> x1(t)=diff(x,t);x2(t)=diff(x1,t);y1(t)=diff(y,t);y2(t)=diff(y1,t);[x0 y0]=dsolve(2x2-x1+9x-(y2+y1+3y)==0,2x2+x1+7x-(y2-y1+5y)0,x(0)1,x1(0)1,y(0)0,y1(0)==0)