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Course Project MAT4109/EDG1109

Project: Mate 4
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%md ## PROBLEM 2 ### POWER SERIES: ### Find the firs 6 nonzero terms in each of two linearly ### independent solutions of the form $\sum{c_nx^n} $, for ### the following differential equation: ### $ xy''+(sin x)y'+xy=0$

PROBLEM 2

POWER SERIES:

Find the firs 6 nonzero terms in each of two linearly

independent solutions of the form cnxn\sum{c_nx^n} , for

the following differential equation:

xy+(sinx)y+xy=0 xy''+(sin x)y'+xy=0

#generating a sum with 12 terms. reset() n=12 a=list(var('a%d'%i)for i in range(n)) x=var('x') y=function('y')(x) var('a0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11') a0=0 a1=1 y(x)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6+a7*x^7+a8*x^8+a9*x^9+a10*x^10+a11*x^11 show(y(x))
(a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11)
a11x11+a10x10+a9x9+a8x8+a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+x\displaystyle a_{11} x^{11} + a_{10} x^{10} + a_{9} x^{9} + a_{8} x^{8} + a_{7} x^{7} + a_{6} x^{6} + a_{5} x^{5} + a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} + x
#expanding the sine function using the McLaurin polynomial to use it in the differential equation sinX=taylor(sin(x),x,0,12) show (sinX) a0=0 a1=1 eq = expand (x*diff(y,x,2)+sinX*diff(y,x,1)+x*y(x)==0) #Substituting the expansion in the differential equation given show(eq)
139916800x11+1362880x915040x7+1120x516x3+x\displaystyle -\frac{1}{39916800} \, x^{11} + \frac{1}{362880} \, x^{9} - \frac{1}{5040} \, x^{7} + \frac{1}{120} \, x^{5} - \frac{1}{6} \, x^{3} + x
x  13628800a11x2113991680a10x20+11362880a11x1914435200a9x19+136288a10x1814989600a8x18115040a11x1715702400a7x17+140320a9x171504a10x1616652800a6x16+145360a8x16+11120a11x1517983360a5x15+151840a7x151560a9x15+112a10x1419979200a4x14+160480a6x141630a8x14116a11x13113305600a3x13+172576a5x131720a7x13+340a9x1353a10x12+a11x12119958400a2x12+190720a4x121840a6x12+115a8x12+a10x11+11a11x11+1120960a3x1111008a5x11+7120a7x1132a9x11+10a10x10+110a11x10+1181440a2x1011260a4x10+120a6x1043a8x10+a9x10139916800x11+90a10x911680a3x9+124a5x976a7x9+a8x9+9a9x912520a2x8+130a4x8a6x8+a7x8+8a8x8+72a9x8+1362880x9+140a3x756a5x7+a6x7+7a7x7+56a8x7+160a2x623a4x6+a5x6+6a6x6+42a7x615040x712a3x5+a4x5+5a5x5+30a6x513a2x4+a3x4+4a4x4+20a5x4+1120x5+a2x3+3a3x3+12a4x3+2a2x2+6a3x216x3+2a2x+x2+x=0\displaystyle x \ {\mapsto}\ -\frac{1}{3628800} \, a_{11} x^{21} - \frac{1}{3991680} \, a_{10} x^{20} + \frac{11}{362880} \, a_{11} x^{19} - \frac{1}{4435200} \, a_{9} x^{19} + \frac{1}{36288} \, a_{10} x^{18} - \frac{1}{4989600} \, a_{8} x^{18} - \frac{11}{5040} \, a_{11} x^{17} - \frac{1}{5702400} \, a_{7} x^{17} + \frac{1}{40320} \, a_{9} x^{17} - \frac{1}{504} \, a_{10} x^{16} - \frac{1}{6652800} \, a_{6} x^{16} + \frac{1}{45360} \, a_{8} x^{16} + \frac{11}{120} \, a_{11} x^{15} - \frac{1}{7983360} \, a_{5} x^{15} + \frac{1}{51840} \, a_{7} x^{15} - \frac{1}{560} \, a_{9} x^{15} + \frac{1}{12} \, a_{10} x^{14} - \frac{1}{9979200} \, a_{4} x^{14} + \frac{1}{60480} \, a_{6} x^{14} - \frac{1}{630} \, a_{8} x^{14} - \frac{11}{6} \, a_{11} x^{13} - \frac{1}{13305600} \, a_{3} x^{13} + \frac{1}{72576} \, a_{5} x^{13} - \frac{1}{720} \, a_{7} x^{13} + \frac{3}{40} \, a_{9} x^{13} - \frac{5}{3} \, a_{10} x^{12} + a_{11} x^{12} - \frac{1}{19958400} \, a_{2} x^{12} + \frac{1}{90720} \, a_{4} x^{12} - \frac{1}{840} \, a_{6} x^{12} + \frac{1}{15} \, a_{8} x^{12} + a_{10} x^{11} + 11 \, a_{11} x^{11} + \frac{1}{120960} \, a_{3} x^{11} - \frac{1}{1008} \, a_{5} x^{11} + \frac{7}{120} \, a_{7} x^{11} - \frac{3}{2} \, a_{9} x^{11} + 10 \, a_{10} x^{10} + 110 \, a_{11} x^{10} + \frac{1}{181440} \, a_{2} x^{10} - \frac{1}{1260} \, a_{4} x^{10} + \frac{1}{20} \, a_{6} x^{10} - \frac{4}{3} \, a_{8} x^{10} + a_{9} x^{10} - \frac{1}{39916800} \, x^{11} + 90 \, a_{10} x^{9} - \frac{1}{1680} \, a_{3} x^{9} + \frac{1}{24} \, a_{5} x^{9} - \frac{7}{6} \, a_{7} x^{9} + a_{8} x^{9} + 9 \, a_{9} x^{9} - \frac{1}{2520} \, a_{2} x^{8} + \frac{1}{30} \, a_{4} x^{8} - a_{6} x^{8} + a_{7} x^{8} + 8 \, a_{8} x^{8} + 72 \, a_{9} x^{8} + \frac{1}{362880} \, x^{9} + \frac{1}{40} \, a_{3} x^{7} - \frac{5}{6} \, a_{5} x^{7} + a_{6} x^{7} + 7 \, a_{7} x^{7} + 56 \, a_{8} x^{7} + \frac{1}{60} \, a_{2} x^{6} - \frac{2}{3} \, a_{4} x^{6} + a_{5} x^{6} + 6 \, a_{6} x^{6} + 42 \, a_{7} x^{6} - \frac{1}{5040} \, x^{7} - \frac{1}{2} \, a_{3} x^{5} + a_{4} x^{5} + 5 \, a_{5} x^{5} + 30 \, a_{6} x^{5} - \frac{1}{3} \, a_{2} x^{4} + a_{3} x^{4} + 4 \, a_{4} x^{4} + 20 \, a_{5} x^{4} + \frac{1}{120} \, x^{5} + a_{2} x^{3} + 3 \, a_{3} x^{3} + 12 \, a_{4} x^{3} + 2 \, a_{2} x^{2} + 6 \, a_{3} x^{2} - \frac{1}{6} \, x^{3} + 2 \, a_{2} x + x^{2} + x = 0
#grouping the coefficients of x and solving the system to find a2 we have : solve ([eq.lhs().coefficient(x,1)==0],a2) show (solve([eq.lhs().coefficient(x,1)==0],a2))
[a2 == (-1/2)]
[a2=(12)\displaystyle a_{2} = \left(-\frac{1}{2}\right)]
#performing this process recursively to find the other coefficients of our series we have : show (solve([eq.lhs().coefficient(x,2).substitute(a2=-1/2)==0],a3))
[a3=0\displaystyle a_{3} = 0]
show (solve([eq.lhs().coefficient(x,3).substitute(a2=-1/2,a3=0)==0],a4))
[a4=(118)\displaystyle a_{4} = \left(\frac{1}{18}\right)]
show (solve([eq.lhs().coefficient(x,4).substitute(a2=-1/2,a3=0,a4=1/18)==0],a5))
[a5=(7360)\displaystyle a_{5} = \left(-\frac{7}{360}\right)]
show (solve([eq.lhs().coefficient(x,5).substitute(a2=-1/2,a3=0,a4=1/18,a5=-7/360)==0],a6))
[a6=(1900)\displaystyle a_{6} = \left(\frac{1}{900}\right)]
show (solve([eq.lhs().coefficient(x,6).substitute(a2=-1/2,a3=0,a4=1/18,a5=-7/360,a6=1/900)==0],a7))
[a7=(157113400)\displaystyle a_{7} = \left(\frac{157}{113400}\right)]
show (solve([eq.lhs().coefficient(x,7).substitute(a2=-1/2,a3=0,a4=1/18,a5=-7/360,a6=1/900,a7=157/113400)==0],a8))
[a8=(1939690)\displaystyle a_{8} = \left(-\frac{19}{39690}\right)]
show (solve([eq.lhs().coefficient(x,8).substitute(a2=-1/2,a3=0,a4=1/18,a5=-7/360,a6=1/900,a7=157/113400,a8=-19/39690)==0],a9))
[a9=(79738102400)\displaystyle a_{9} = \left(\frac{797}{38102400}\right)]
show (solve([eq.lhs().coefficient(x,9).substitute(a2=-1/2,a3=0,a4=1/18,a5=-7/360,a6=1/900,a7=157/113400,a8=-19/39690,a9=797/38102400)==0],a10))
[a10=(92330618000)\displaystyle a_{10} = \left(\frac{923}{30618000}\right)]
show (solve([eq.lhs().coefficient(x,10).substitute(a2=-1/2,a3=0,a4=1/18,a5=-7/360,a6=1/900,a7=157/113400,a8=-19/39690,a9=797/38102400,a10=923/30618000)==0],a11))
[a11=(41551947151720000)\displaystyle a_{11} = \left(-\frac{415519}{47151720000}\right)]
#Now that we have calculated the coefficients we can calculate our first solution y1 (x)=0+1*x+(-1/2)*x^2+(0)*x^3+(1/18)*x^4+(-7/360)*x^5+(1/900)*x^6+(157/113400)*x^7+(-19/39690)*x^8+(797/38102400)*x^9+(923/30618000)*x^10+(-415519/47151720000)*x^11 show (y1(x))
41551947151720000x11+92330618000x10+79738102400x91939690x8+157113400x7+1900x67360x5+118x412x2+x\displaystyle -\frac{415519}{47151720000} \, x^{11} + \frac{923}{30618000} \, x^{10} + \frac{797}{38102400} \, x^{9} - \frac{19}{39690} \, x^{8} + \frac{157}{113400} \, x^{7} + \frac{1}{900} \, x^{6} - \frac{7}{360} \, x^{5} + \frac{1}{18} \, x^{4} - \frac{1}{2} \, x^{2} + x
#now we do the analogous process for our second linearly independent solution #generating a sum with 12 terms . reset () n=12 a= list (var('a%d'%i) for i in range (n)) x=var('x') y=function('y')(x) var('a0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11') a0=1 a1=0 y(x)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6+a7*x^7+a8*x^8+a9*x^9+a10*x^10+a11*x^11 show(y(x))
(a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11)
a11x11+a10x10+a9x9+a8x8+a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+1\displaystyle a_{11} x^{11} + a_{10} x^{10} + a_{9} x^{9} + a_{8} x^{8} + a_{7} x^{7} + a_{6} x^{6} + a_{5} x^{5} + a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} + 1
#expanding the sine function using the Mclaurin polynomial to use it in the differential equation senoX=taylor(sin(x),x,0,12) show (senoX) a0=1 a1=0 eq = expand (x*diff(y,x,2)+senoX*diff(y,x,1)+x*y(x)==0) # Substituting the expansion in the differential equation given show(eq)
139916800x11+1362880x915040x7+1120x516x3+x\displaystyle -\frac{1}{39916800} \, x^{11} + \frac{1}{362880} \, x^{9} - \frac{1}{5040} \, x^{7} + \frac{1}{120} \, x^{5} - \frac{1}{6} \, x^{3} + x
x  13628800a11x2113991680a10x20+11362880a11x1914435200a9x19+136288a10x1814989600a8x18115040a11x1715702400a7x17+140320a9x171504a10x1616652800a6x16+145360a8x16+11120a11x1517983360a5x15+151840a7x151560a9x15+112a10x1419979200a4x14+160480a6x141630a8x14116a11x13113305600a3x13+172576a5x131720a7x13+340a9x1353a10x12+a11x12119958400a2x12+190720a4x121840a6x12+115a8x12+a10x11+11a11x11+1120960a3x1111008a5x11+7120a7x1132a9x11+10a10x10+110a11x10+1181440a2x1011260a4x10+120a6x1043a8x10+a9x10+90a10x911680a3x9+124a5x976a7x9+a8x9+9a9x912520a2x8+130a4x8a6x8+a7x8+8a8x8+72a9x8+140a3x756a5x7+a6x7+7a7x7+56a8x7+160a2x623a4x6+a5x6+6a6x6+42a7x612a3x5+a4x5+5a5x5+30a6x513a2x4+a3x4+4a4x4+20a5x4+a2x3+3a3x3+12a4x3+2a2x2+6a3x2+2a2x+x=0\displaystyle x \ {\mapsto}\ -\frac{1}{3628800} \, a_{11} x^{21} - \frac{1}{3991680} \, a_{10} x^{20} + \frac{11}{362880} \, a_{11} x^{19} - \frac{1}{4435200} \, a_{9} x^{19} + \frac{1}{36288} \, a_{10} x^{18} - \frac{1}{4989600} \, a_{8} x^{18} - \frac{11}{5040} \, a_{11} x^{17} - \frac{1}{5702400} \, a_{7} x^{17} + \frac{1}{40320} \, a_{9} x^{17} - \frac{1}{504} \, a_{10} x^{16} - \frac{1}{6652800} \, a_{6} x^{16} + \frac{1}{45360} \, a_{8} x^{16} + \frac{11}{120} \, a_{11} x^{15} - \frac{1}{7983360} \, a_{5} x^{15} + \frac{1}{51840} \, a_{7} x^{15} - \frac{1}{560} \, a_{9} x^{15} + \frac{1}{12} \, a_{10} x^{14} - \frac{1}{9979200} \, a_{4} x^{14} + \frac{1}{60480} \, a_{6} x^{14} - \frac{1}{630} \, a_{8} x^{14} - \frac{11}{6} \, a_{11} x^{13} - \frac{1}{13305600} \, a_{3} x^{13} + \frac{1}{72576} \, a_{5} x^{13} - \frac{1}{720} \, a_{7} x^{13} + \frac{3}{40} \, a_{9} x^{13} - \frac{5}{3} \, a_{10} x^{12} + a_{11} x^{12} - \frac{1}{19958400} \, a_{2} x^{12} + \frac{1}{90720} \, a_{4} x^{12} - \frac{1}{840} \, a_{6} x^{12} + \frac{1}{15} \, a_{8} x^{12} + a_{10} x^{11} + 11 \, a_{11} x^{11} + \frac{1}{120960} \, a_{3} x^{11} - \frac{1}{1008} \, a_{5} x^{11} + \frac{7}{120} \, a_{7} x^{11} - \frac{3}{2} \, a_{9} x^{11} + 10 \, a_{10} x^{10} + 110 \, a_{11} x^{10} + \frac{1}{181440} \, a_{2} x^{10} - \frac{1}{1260} \, a_{4} x^{10} + \frac{1}{20} \, a_{6} x^{10} - \frac{4}{3} \, a_{8} x^{10} + a_{9} x^{10} + 90 \, a_{10} x^{9} - \frac{1}{1680} \, a_{3} x^{9} + \frac{1}{24} \, a_{5} x^{9} - \frac{7}{6} \, a_{7} x^{9} + a_{8} x^{9} + 9 \, a_{9} x^{9} - \frac{1}{2520} \, a_{2} x^{8} + \frac{1}{30} \, a_{4} x^{8} - a_{6} x^{8} + a_{7} x^{8} + 8 \, a_{8} x^{8} + 72 \, a_{9} x^{8} + \frac{1}{40} \, a_{3} x^{7} - \frac{5}{6} \, a_{5} x^{7} + a_{6} x^{7} + 7 \, a_{7} x^{7} + 56 \, a_{8} x^{7} + \frac{1}{60} \, a_{2} x^{6} - \frac{2}{3} \, a_{4} x^{6} + a_{5} x^{6} + 6 \, a_{6} x^{6} + 42 \, a_{7} x^{6} - \frac{1}{2} \, a_{3} x^{5} + a_{4} x^{5} + 5 \, a_{5} x^{5} + 30 \, a_{6} x^{5} - \frac{1}{3} \, a_{2} x^{4} + a_{3} x^{4} + 4 \, a_{4} x^{4} + 20 \, a_{5} x^{4} + a_{2} x^{3} + 3 \, a_{3} x^{3} + 12 \, a_{4} x^{3} + 2 \, a_{2} x^{2} + 6 \, a_{3} x^{2} + 2 \, a_{2} x + x = 0
solve([eq.lhs().coefficient(x,1)==0],a2) show(solve([eq.lhs().coefficient(x,1)==0],a2))
[a2 == (-1/2)]
[a2=(12)\displaystyle a_{2} = \left(-\frac{1}{2}\right)]
show(solve([eq.lhs().coefficient(x,2).substitute(a2=-1/2)==0],a3))
[a3=(16)\displaystyle a_{3} = \left(\frac{1}{6}\right)]
show(solve([eq.lhs().coefficient(x,3).substitute(a2=-1/2,a3=1/6)==0],a4))
[a4=0\displaystyle a_{4} = 0]
show(solve([eq.lhs().coefficient(x,4).substitute(a2=-1/2,a3=1/6,a4=0)==0],a5))
[a5=(160)\displaystyle a_{5} = \left(-\frac{1}{60}\right)]
show(solve([eq.lhs().coefficient(x,5).substitute(a2=-1/2,a3=1/6,a4=0,a5=-1/60)==0],a6))
[a6=(1180)\displaystyle a_{6} = \left(\frac{1}{180}\right)]
show(solve([eq.lhs().coefficient(x,6).substitute(a2=-1/2,a3=1/6,a4=0,a5=-1/60,a6=1/180)==0],a7))
[a7=(15040)\displaystyle a_{7} = \left(-\frac{1}{5040}\right)]
show(solve([eq.lhs().coefficient(x,7).substitute(a2=-1/2,a3=1/6,a4=0,a5=-1/60,a6=1/180,a7=-1/5040)==0],a8))
[a8=(12520)\displaystyle a_{8} = \left(-\frac{1}{2520}\right)]
show(solve([eq.lhs().coefficient(x,8).substitute(a2=-1/2,a3 =1/6,a4=0,a5=-1/60,a6=1/180,a7=-1/5040,a8=-1/2520)==0],a9))
[a9=(1190720)\displaystyle a_{9} = \left(\frac{11}{90720}\right)]
show(solve([eq.lhs().coefficient(x,9).substitute(a2=-1/2,a3 =1/6,a4=0,a5=-1/60,a6=1/180,a7=-1/5040,a8=-1/2520,a9=11/90720)==0],a10))
[a10=(1680400)\displaystyle a_{10} = \left(-\frac{1}{680400}\right)]
show(solve([eq.lhs().coefficient(x,10).substitute(a2=-1/2,a3=1/6,a4=0,a5=-1/60,a6=1/180,a7=-1/5040,a8=-1/2520,a9=11/90720,a10=-1/680400)==0],a11))
[a11=(4957598752000)\displaystyle a_{11} = \left(-\frac{4957}{598752000}\right)]
y2(x)=1+0*x+(-1/2)*x^2+(1/6)*x^3+(0)*x^4+(-1/60)*x^5+(1/180)*x^6+(-1/5040)*x^7+(-1/2520)*x^8+(11/90720)*x^9+(-1/680400)*x^10+(-4957/598752000)*x^11 show(y2(x))
4957598752000x111680400x10+1190720x912520x815040x7+1180x6160x5+16x312x2+1\displaystyle -\frac{4957}{598752000} \, x^{11} - \frac{1}{680400} \, x^{10} + \frac{11}{90720} \, x^{9} - \frac{1}{2520} \, x^{8} - \frac{1}{5040} \, x^{7} + \frac{1}{180} \, x^{6} - \frac{1}{60} \, x^{5} + \frac{1}{6} \, x^{3} - \frac{1}{2} \, x^{2} + 1