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 = 0.251201 (sec), leaf count = 47
\[\left \{\left \{y(x)\to \frac {4 a^4 \left (\left (c_4 x+c_3\right ) \sin (a x)+\left (c_2 x+c_1\right ) \cos (a x)\right )+\cosh (a x)}{4 a^4}\right \}\right \}\]
Maple ✓
cpu = 0.496 (sec), leaf count = 51
\[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{-ax}}+ \left ( 8\,{\it \_C3}\,x+8\,{\it \_C1} \right ) {a}^{4}\cos \left ( ax \right ) + \left ( 8\,{\it \_C4}\,x+8\,{\it \_C2} \right ) {a}^{4}\sin \left ( ax \right ) +{{\rm e}^{ax}}}{8\,{a}^{4}}} \right \} \] 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] -> (Cosh[a*x] + 4*a^4*((C[1] + x*C[2])*Cos[a*x] + (C[3] + x*C[4])*Sin[a*x]))/(4*a^4)}}
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),'implicit')
Maple raw output
y(x) = 1/8*(exp(-a*x)+(8*_C3*x+8*_C1)*a^4*cos(a*x)+(8*_C4*x+8*_C2)*a^4*sin(a*x)+exp(a*x))/a^4