60.6.8 problem 1585
Internal
problem
ID
[11584]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
5,
linear
fifth
and
higher
order
Problem
number
:
1585
Date
solved
:
Monday, January 27, 2025 at 11:23:43 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} x \left (a y^{\prime }+b y^{\prime \prime }+c y^{\prime \prime \prime }+e y^{\prime \prime \prime \prime }\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 673
dsolve(x * (a*diff(y(x),x) + b*diff(y(x),x$2) + c*diff(y(x),x$3) + e*diff(y(x),x$4))*y(x)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= {\mathrm e}^{\frac {\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{2}/{3}}-2 c \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{1}/{3}}-12 b e +4 c^{2}\right ) x}{6 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{1}/{3}}}} c_4 +c_3 \,{\mathrm e}^{\frac {x \left (i \left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{2}/{3}}+12 b e -4 c^{2}\right ) \sqrt {3}+12 b e -{\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{1}/{3}}+2 c \right )}^{2}\right )}{12 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{1}/{3}}}}+c_{2} {\mathrm e}^{\frac {\left (-i \left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{2}/{3}}+12 b e -4 c^{2}\right ) \sqrt {3}+12 b e -{\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{1}/{3}}+2 c \right )}^{2}\right ) x}{12 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a b c +4 b^{3}\right ) e +4 c^{3} a -b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{{1}/{3}}}}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 132
DSolve[x * (a*D[y[x],x] + b*D[y[x],{x,2}] + c*D[y[x],{x,3}] + e*D[y[x],{x,4}])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to 0 \\
y(x)\to \int _1^x\left (e^{K[1] \text {Root}\left [\text {$\#$1}^3+\frac {c \text {$\#$1}^2}{e}+\frac {b \text {$\#$1}}{e}+\frac {a}{e}\&,1\right ]} c_1+e^{K[1] \text {Root}\left [\text {$\#$1}^3+\frac {c \text {$\#$1}^2}{e}+\frac {b \text {$\#$1}}{e}+\frac {a}{e}\&,2\right ]} c_2+e^{K[1] \text {Root}\left [\text {$\#$1}^3+\frac {c \text {$\#$1}^2}{e}+\frac {b \text {$\#$1}}{e}+\frac {a}{e}\&,3\right ]} c_3\right )dK[1]+c_4 \\
\end{align*}