Please explain the last step of how to get the general solution:\ \ (a)

y^(10)-4y^(9)+11y^(8)+8y^(7)-25y^(6)-4y^(5)+13y^(4)=0

\ The auxilary equation is

m^(10)-4m^(9)+11m^(8)+8m^(7)-25m^(6)-4m^(5)+13m^(4)=0

.\ Solve this equation:\

m^(10)-4m^(9)+11m^(8)+8m^(7)-25m^(6)-4m^(5)+13m^(4)=0\ m^(4)(m^(6)-4m^(5)+11m^(4)+8m^(3)-25m^(2)-4m+13)=0\ m^(4)(m^(2)-4m+13)(m-1)^(2)(m+1)^(2)=0\ m=0,0,0,0,1,1,-1,-1,2-3i,2+3i.

\ So, the general solution is\

y(x)=(c_(1)+c_(2)x+c_(3)x^(2)+c_(4)x^(3))e^(0x)+(c_(5)+c_(6)x)e^(x)+(c_(7)+c_(8)x)e^(-x)+c_(0)e^(2x)cos3x+c_(10)e^(2x)sin3x

(a)y104y9+11y8+8y725y64y5+13y4=0 The auxilary equation is m104m9+11m8+8m725m64m5+13m4=0. Solve this equation: m104m9+11m8+8m725m64m5+13m4=0m4(m64m5+11m4+8m325m24m+13)=0m4(m24m+13)(m1)2(m+1)2=0m=0,0,0,0,1,1,1,1,23i,2+3i. So, the general solution is y(x)=(c1+c2x+c3x2+c4x3)e0x+(c5+c6x)ex+(c7+c8x)ex+c0e2xcos3x+c10e2xsin3 (a)y104y9+11y8+8y725y64y5+13y4=0 The auxilary equation is m104m9+11m8+8m725m64m5+13m4=0. Solve this equation: m104m9+11m8+8m725m64m5+13m4=0m4(m64m5+11m4+8m325m24m+13)=0m4(m24m+13)(m1)2(m+1)2=0m=0,0,0,0,1,1,1,1,23i,2+3i. So, the general solution is y(x)=(c1+c2x+c3x2+c4x3)e0x+(c5+c6x)ex+(c7+c8x)ex+c0e2xcos3x+c10e2xsin3