ODE
\[ y''''(x)+\left (a^2+b^2\right ) y''(x)+a^2 b^2 y(x)=0 \] ODE Classification
[[_high_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0086521 (sec), leaf count = 34
\[\left \{\left \{y(x)\to c_4 \sin (a x)+c_3 \cos (a x)+c_2 \sin (b x)+c_1 \cos (b x)\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 29
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sin \left ( ax \right ) +{\it \_C2}\,\cos \left ( ax \right ) +{\it \_C3}\,\sin \left ( bx \right ) +{\it \_C4}\,\cos \left ( bx \right ) \right \} \] Mathematica raw input
DSolve[a^2*b^2*y[x] + (a^2 + b^2)*y''[x] + y''''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[3]*Cos[a*x] + C[1]*Cos[b*x] + C[4]*Sin[a*x] + C[2]*Sin[b*x]}}
Maple raw input
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+(a^2+b^2)*diff(diff(y(x),x),x)+a^2*b^2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*sin(a*x)+_C2*cos(a*x)+_C3*sin(b*x)+_C4*cos(b*x)