Internal problem ID [9147]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1568.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime } x^{3}+12 x^{2} y^{\prime \prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 89
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+8*x^3*diff(diff(diff(y(x),x),x),x)+12*x^2*diff(diff(y(x),x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {1}{2}-\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{2} x^{-\frac {1}{2}+\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{3} x^{-\frac {1}{2}-\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}+c_{4} x^{-\frac {1}{2}+\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 113
DSolve[a*y[x] + 12*x^2*y''[x] + 8*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_3 x^{-\frac {1}{2} \sqrt {4 \sqrt {1-a}+5}}+c_4 x^{\frac {1}{2} \sqrt {4 \sqrt {1-a}+5}}+x^{-\frac {1}{2} \sqrt {5-4 \sqrt {1-a}}} \left (c_2 x^{\sqrt {5-4 \sqrt {1-a}}}+c_1\right )}{\sqrt {x}} \\ \end{align*}