60.5.6 problem 1539

Internal problem ID [11540]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1539
Date solved : Monday, January 27, 2025 at 11:23:19 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+\left (\lambda +1\right ) a^{2} y^{\prime \prime }+\lambda \,a^{4} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 35

dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+(lambda+1)*a^2*diff(diff(y(x),x),x)+lambda*a^4*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \sin \left (a x \right )+c_{2} \cos \left (a x \right )+c_3 \sin \left (a \sqrt {\lambda }\, x \right )+c_4 \cos \left (a \sqrt {\lambda }\, x \right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 44

DSolve[a^4*\[Lambda]*y[x] + a^2*(1 + \[Lambda])*D[y[x],{x,2}] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \cos \left (a \sqrt {\lambda } x\right )+c_2 \sin \left (a \sqrt {\lambda } x\right )+c_3 \cos (a x)+c_4 \sin (a x) \]