60.6.3 problem 1580
Internal
problem
ID
[11579]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
5,
linear
fifth
and
higher
order
Problem
number
:
1580
Date
solved
:
Monday, January 27, 2025 at 11:23:38 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\left (6\right )}+y-\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 71
dsolve(diff(y(x),x$6)+y(x)-sin(3/2*x)*sin(1/2*x)=0,y(x), singsol=all)
\[
y = \left (c_4 \sin \left (\frac {x}{2}\right )+c_3 \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (c_6 \sin \left (\frac {x}{2}\right )+c_5 \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+\frac {\cos \left (2 x \right )}{126}+\frac {\left (5+24 c_{1} \right ) \cos \left (x \right )}{24}+\frac {\sin \left (x \right ) \left (x +12 c_{2} \right )}{12}
\]
✓ Solution by Mathematica
Time used: 2.885 (sec). Leaf size: 498
DSolve[D[y[x],{x,6}]+y[x]-Sin[3/2*x]*Sin[1/2*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to e^{-\frac {\sqrt {3} x}{2}} \left (e^{\frac {\sqrt {3} x}{2}} \cos (x) \int _1^x-\frac {2}{3} \left (2 \cos \left (\frac {K[2]}{2}\right )+\cos \left (\frac {3 K[2]}{2}\right )\right ) \sin ^3\left (\frac {K[2]}{2}\right )dK[2]+e^{\sqrt {3} x} \cos \left (\frac {x}{2}\right ) \int _1^x-\frac {1}{6} e^{-\frac {1}{2} \sqrt {3} K[1]} (2 \cos (K[1])+1) \sin ^2\left (\frac {K[1]}{2}\right ) \left (\sqrt {3} \cos \left (\frac {K[1]}{2}\right )+\sin \left (\frac {K[1]}{2}\right )\right )dK[1]+\cos \left (\frac {x}{2}\right ) \int _1^x\frac {1}{6} e^{\frac {1}{2} \sqrt {3} K[3]} (2 \cos (K[3])+1) \left (\sqrt {3} \cos \left (\frac {K[3]}{2}\right )-\sin \left (\frac {K[3]}{2}\right )\right ) \sin ^2\left (\frac {K[3]}{2}\right )dK[3]+\sin \left (\frac {x}{2}\right ) \int _1^x\frac {1}{6} e^{\frac {1}{2} \sqrt {3} K[4]} (2 \cos (K[4])+1) \sin ^2\left (\frac {K[4]}{2}\right ) \left (\cos \left (\frac {K[4]}{2}\right )+\sqrt {3} \sin \left (\frac {K[4]}{2}\right )\right )dK[4]+e^{\frac {\sqrt {3} x}{2}} \sin (x) \int _1^x\frac {1}{3} (\cos (K[5])+\cos (2 K[5])+1) \sin ^2\left (\frac {K[5]}{2}\right )dK[5]+e^{\sqrt {3} x} \sin \left (\frac {x}{2}\right ) \int _1^x-\frac {1}{6} e^{-\frac {1}{2} \sqrt {3} K[6]} (2 \cos (K[6])+1) \sin ^2\left (\frac {K[6]}{2}\right ) \left (\sqrt {3} \sin \left (\frac {K[6]}{2}\right )-\cos \left (\frac {K[6]}{2}\right )\right )dK[6]+c_1 e^{\sqrt {3} x} \cos \left (\frac {x}{2}\right )+c_3 \cos \left (\frac {x}{2}\right )+c_2 e^{\frac {\sqrt {3} x}{2}} \cos (x)+c_4 \sin \left (\frac {x}{2}\right )+c_6 e^{\sqrt {3} x} \sin \left (\frac {x}{2}\right )+c_5 e^{\frac {\sqrt {3} x}{2}} \sin (x)\right )
\]