Internal problem ID [9117]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1538.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y-\cosh \left (a x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.415 (sec). Leaf size: 48
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)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{-a x} \left (1+{\mathrm e}^{2 a x}\right )}{8 a^{4}}+c_{1} \cos \left (a x \right )+c_{2} \sin \left (a x \right )+c_{3} \cos \left (a x \right ) x +c_{4} \sin \left (a x \right ) x \]
✓ Solution by Mathematica
Time used: 0.118 (sec). Leaf size: 41
DSolve[-Cosh[a*x] + a^4*y[x] + 2*a^2*y''[x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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) \\ \end{align*}