Answer
(a) $y'' = m^2y$
(b) $y = 2~sinh~3x-4~cosh~3x$
Work Step by Step
(a) $y = A~sinh~mx+B~cosh~mx$
$y' = mA~cosh~mx+mB~sinh~mx$
$y'' = m^2A~sinh~mx+m^2B~cosh~mx = m^2y$
(b) $y'' = m^2y$
If $y'' = 9y,$ then $m^2=9$ and $m = 3$
If $y(0) = -4$:
$A~sinh~m(0)+B~cosh~m(0) = -4$
$0+B~(1) = -4$
$B = -4$
If $y'(0) = 6$:
$mA~cosh~m(0)+mB~sinh~m(0) = 6$
$mA(1)+B~(0) = 6$
$mA = 6$
$A = \frac{6}{3}$
$A = 2$
Therefore:
$y = 2~sinh~3x-4~cosh~3x$