4.45.31 $$y''''(x)+a^4 y(x)+2 a^2 y''(x)=\cosh (a x)$$

ODE
$y''''(x)+a^4 y(x)+2 a^2 y''(x)=\cosh (a x)$ ODE Classiﬁcation

[[_high_order, _linear, _nonhomogeneous]]

Book solution method
TO DO

Mathematica
cpu = 0.309896 (sec), leaf count = 41

$\left \{\left \{y(x)\to \frac {\cosh (a x)}{4 a^4}+(c_2 x+c_1) \cos (a x)+(c_4 x+c_3) \sin (a x)\right \}\right \}$

Maple
cpu = 5.768 (sec), leaf count = 48

$\left [y \left (x \right ) = \frac {{\mathrm e}^{-a x} \left (1+{\mathrm e}^{2 a x}\right )}{8 a^{4}}+\cos \left (a x \right ) \textit {\_C1} +\sin \left (a x \right ) \textit {\_C2} +\textit {\_C3} \cos \left (a x \right ) x +\textit {\_C4} \sin \left (a x \right ) x\right ]$ Mathematica raw input

DSolve[a^4*y[x] + 2*a^2*y''[x] + y''''[x] == Cosh[a*x],y[x],x]

Mathematica raw output

{{y[x] -> (C[1] + x*C[2])*Cos[a*x] + Cosh[a*x]/(4*a^4) + (C[3] + x*C[4])*Sin[a*x
]}}

Maple raw input

dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+2*a^2*diff(diff(y(x),x),x)+a^4*y(x) = cosh(a*x), y(x))

Maple raw output

[y(x) = 1/8*exp(-a*x)*(1+exp(2*a*x))/a^4+cos(a*x)*_C1+sin(a*x)*_C2+_C3*cos(a*x)*
x+_C4*sin(a*x)*x]