Question

solve the following higher order differential equation?

y^(6)-64y=0

Answer 1

A fairly simple homogeneous equation. Solve the following characteristic equation:
\[r^6-64=0\]
Note that \(2^6=64\), so you have a difference of the form \(a^6-b^6\), which can be expressed as the difference of cubes \(\left(c^2\right)^3-\left(d^2\right)^3\).

Basically, what I mean by all that, is that you can factor easily:
\[r^6-64=0\\
\left(r^2\right)^3-\left(4\right)^3=0\\
\left(r^2-4\right)\left(\left(r^2\right)^2+4r^2+4^2\right)=0\\
(r+2)(r-2)\left(r^4+4r^2+16\right)=0\]
From here, solve for \(r\).

Answer 2

A minor detail I missed above:
\[(r+2)(r-2)\left(r^4+4r^2+16\right)=0\]
factors further to get
\[(r+2)(r-2)(r^2+2r+4)(r^2-2r+4)=0\]

Answer 3

Thank you so much,I'am check your solution it's right

Answer 4

You're welcome!