r^n*(2/pi*(8*n^2*pi^2*sin(pi*n)*cos(pi*n)-4*sin(pi*n)*cos(pi*n)+8*pi*n*cos(pi*n)^2-4*pi*n+3*n*cos(pi*n)^2-3*n+6*pi*n^2*sin(pi*n)*cos(pi*n)+5*n^2*sin(pi*n)*cos(pi*n))/n^3*cos(n*x)+2/pi*(-8*n^2*pi^2*cos(pi*n)^2+4*n^2*pi^2+4*cos(pi*n)^2-4+8*pi*n*sin(pi*n)*cos(pi*n)+3*n*sin(pi*n)*cos(pi*n)-6*pi*n^2*cos(pi*n)^2+3*pi*n^2-5*n^2*cos(pi*n)^2+5*n^2)/n^3*sin(n*x))
u1=subs(un, n,1)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u1 =
r*(8*cos(x)+2/pi*(-4*pi^2-3*pi)*sin(x))
u2=subs(un, n, 2)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u2 =
r^2*(2*cos(2*x)+1/4/pi*(-16*pi^2-12*pi)*sin(2*x))
u3=subs(un, n, 3)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u3 =
r^3*(8/9*cos(3*x)+2/27/pi*(-36*pi^2-27*pi)*sin(3*x))
u4=subs(un, n, 4)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u4 =
r^4*(1/2*cos(4*x)+1/32/pi*(-64*pi^2-48*pi)*sin(4*x))
u5=subs(un, n, 5)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u5 =
r^5*(8/25*cos(5*x)+2/125/pi*(-100*pi^2-75*pi)*sin(5*x))
u6=subs(un, n, 6)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u6 =
r^6*(2/9*cos(6*x)+1/108/pi*(-144*pi^2-108*pi)*sin(6*x))
u7=subs(un, n, 7)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u7 =
r^7*(8/49*cos(7*x)+2/343/pi*(-196*pi^2-147*pi)*sin(7*x))
u8=subs(un, n, 8)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u8 =
r^8*(1/8*cos(8*x)+1/256/pi*(-256*pi^2-192*pi)*sin(8*x))
u9=subs(un, n, 9)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u9 =
r^9*(8/81*cos(9*x)+2/729/pi*(-324*pi^2-243*pi)*sin(9*x))
u10=subs(un, n, 10)
Warning: Reference to uninitialized variable r in sym/subs at line 118.
> In C:\PROGRAMM\MATLAB6p1\toolbox\symbolic\@sym\subs.m at line 118
u10 =
r^10*(2/25*cos(10*x)+1/500/pi*(-400*pi^2-300*pi)*sin(10*x))
Общее решение:
u=simple(A0+u1+u2+u3+u4+u5+u6+u7+u8+u9+u10)
u =
28670805696113295/18014398509481984*pi-8*r*sin(x)*pi+17202483417667977/18014398509481984*pi^2+1911387046407553/2251799813685248*pi^3-3/5*r^10*sin(10*x)+2/25*r^10*cos(10*x)-2/3*r^9*sin(9*x)+8/81*r^9*cos(9*x)-3/4*r^8*sin(8*x)+1/8*r^8*cos(8*x)-6/7*r^7*sin(7*x)+8/49*r^7*cos(7*x)-r^6*sin(6*x)+2/9*r^6*cos(6*x)-6/5*r^5*sin(5*x)+8/25*r^5*cos(5*x)-3/2*r^4*sin(4*x)+1/2*r^4*cos(4*x)-2*r^3*sin(3*x)+8/9*r^3*cos(3*x)-3*r^2*sin(2*x)+2*r^2*cos(2*x)-6*r*sin(x)+8*r*cos(x)-4*r^2*sin(2*x)*pi-8/3*r^3*sin(3*x)*pi-2*r^4*sin(4*x)*pi-8/5*r^5*sin(5*x)*pi-4/3*r^6*sin(6*x)*pi-8/7*r^7*sin(7*x)*pi-r^8*sin(8*x)*pi-8/9*r^9*sin(9*x)*pi-4/5*r^10*sin(10*x)*pi
Проверка:
simplify(1/r*diff(r*diff(u, 'r'), 'r')+1/(r*r)*diff(u, 2, 'x'))
ans =
0
Подставляем граничные условия:
uu=simplify(subs(u, r, R))
uu =
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