Find the general solution of the given higher-order differential equation.

d ^{4}y |

dx^{4} |

− 2

d ^{2}y |

dx^{2} |

− 8y = 0

y(x) =

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(d^(4)y)/(dx^(4))-2(d^(2)y)/(dx^(2))-8y=0The characteristic equation associated to the given homogeneous equation is:{:[r^(4)-2r^(2)-8=0],[=>r^(4)-4r^(2)+2r^(2)-8=0],[=>r^(2)(r^(2)-4)+2(r^(2)-4)=0],[=>(r^(2)+2)(r^(2)-4)=0],[=>(r^(2)+2)(r-2)(r+2)=0],[=>r^(2)+2=0quad" or, "],[r-2=0],[r+2=0],[r" or "],[r=2],[" or "quad r=2],[" or "quad r=2]:}So, The general solution of the Differential equation is:y(x)=c_(1)e^(-2x)+c_(2)e^(2x)+c_(3)cos(sqrt2x)+c_(4)sin(sqrt2x)Where,C_(1),C_(2),C_(3),C_(4) are arbitrary constants Answer) ...