6.3 problem 1580

Internal problem ID [9159]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1580.
ODE order: 6.
ODE degree: 1.

CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\relax (6)}+y-\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )=0} \end {gather*}

Solution by Maple

Time used: 0.093 (sec). Leaf size: 80

dsolve(diff(y(x),x$6)+y(x)-sin(3/2*x)*sin(1/2*x)=0,y(x), singsol=all)
 

\[ y \relax (x ) = \frac {5 \cos \relax (x )}{24}+\frac {\cos \left (2 x \right )}{126}+\frac {x \sin \relax (x )}{12}+\cos \relax (x ) c_{1}+\sin \relax (x ) c_{2}+c_{3} {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}} \cos \left (\frac {x}{2}\right )+c_{4} {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}} \sin \left (\frac {x}{2}\right )+c_{5} {\mathrm e}^{\frac {\sqrt {3}\, x}{2}} \cos \left (\frac {x}{2}\right )+c_{6} {\mathrm e}^{\frac {\sqrt {3}\, x}{2}} \sin \left (\frac {x}{2}\right ) \]

Solution by Mathematica

Time used: 1.146 (sec). Leaf size: 93

DSolve[y''''''[x]+y[x]-Sin[3/2*x]*Sin[1/2*x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{12} x \sin (x)+\frac {1}{126} \cos (2 x)+\left (\frac {1}{4}+c_2\right ) \cos (x)+c_5 \sin (x)+e^{-\frac {\sqrt {3} x}{2}} \left (\left (c_1 e^{\sqrt {3} x}+c_3\right ) \cos \left (\frac {x}{2}\right )+\left (c_6 e^{\sqrt {3} x}+c_4\right ) \sin \left (\frac {x}{2}\right )\right ) \\ \end{align*}