Internal problem ID [9908]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1585.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _missing_x]]
Solve \begin {gather*} \boxed {x \left (y^{\prime } a +b y^{\prime \prime }+c y^{\prime \prime \prime }+e y^{\prime \prime \prime \prime }\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 675
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 \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\frac {\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {2}{3}}-2 c \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {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 c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}}} c_{4} +{\mathrm e}^{\frac {x \left (i \left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {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 c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}+2 c \right )}^{2}\right )}{12 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}}} c_{3} +c_{2} {\mathrm e}^{\frac {\left (-i \left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {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 c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}+2 c \right )}^{2}\right ) x}{12 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}}}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.173 (sec). Leaf size: 214
DSolve[x * (a*y'[x] + b*y''[x] + c*y'''[x] + e*y''''[x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 0 \\ y(x)\to \frac {c_1 e^{x \text {Root}\left [\text {$\#$1}^3+\frac {\text {$\#$1}^2 c}{e}+\frac {\text {$\#$1} b}{e}+\frac {a}{e}\&,1\right ]}}{\text {Root}\left [\text {$\#$1}^3+\frac {\text {$\#$1}^2 c}{e}+\frac {\text {$\#$1} b}{e}+\frac {a}{e}\&,1\right ]}+\frac {c_2 e^{x \text {Root}\left [\text {$\#$1}^3+\frac {\text {$\#$1}^2 c}{e}+\frac {\text {$\#$1} b}{e}+\frac {a}{e}\&,2\right ]}}{\text {Root}\left [\text {$\#$1}^3+\frac {\text {$\#$1}^2 c}{e}+\frac {\text {$\#$1} b}{e}+\frac {a}{e}\&,2\right ]}+\frac {c_3 e^{x \text {Root}\left [\text {$\#$1}^3+\frac {\text {$\#$1}^2 c}{e}+\frac {\text {$\#$1} b}{e}+\frac {a}{e}\&,3\right ]}}{\text {Root}\left [\text {$\#$1}^3+\frac {\text {$\#$1}^2 c}{e}+\frac {\text {$\#$1} b}{e}+\frac {a}{e}\&,3\right ]}+c_4 \\ \end{align*}