60.5.5 problem 1538

Internal problem ID [11539]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1538
Date solved : Monday, January 27, 2025 at 11:23:18 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y-\cosh \left (a x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 51

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)
 

\[ y = \frac {{\mathrm e}^{-a x}+\left (8 x c_3 +8 c_{1} \right ) a^{4} \cos \left (a x \right )+\left (8 c_4 x +8 c_{2} \right ) a^{4} \sin \left (a x \right )+{\mathrm e}^{a x}}{8 a^{4}} \]

✓ Solution by Mathematica

Time used: 0.293 (sec). Leaf size: 176

DSolve[-Cosh[a*x] + a^4*y[x] + 2*a^2*D[y[x],{x,2}] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos (a x) \int _1^x\frac {\cosh (a K[1]) (a \cos (a K[1]) K[1]-\sin (a K[1]))}{2 a^3}dK[1]+\sin (a x) \int _1^x\frac {\cosh (a K[3]) (\cos (a K[3])+a K[3] \sin (a K[3]))}{2 a^3}dK[3]+x \cos (a x) \int _1^x-\frac {\cos (a K[2]) \cosh (a K[2])}{2 a^2}dK[2]+x \sin (a x) \int _1^x-\frac {\cosh (a K[4]) \sin (a K[4])}{2 a^2}dK[4]+c_1 \cos (a x)+c_2 x \cos (a x)+c_3 \sin (a x)+c_4 x \sin (a x) \]