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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:
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
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