Internal problem ID [9059]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1480.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (x -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 35
dsolve(x*diff(diff(diff(y(x),x),x),x)-(x+2*v)*diff(diff(y(x),x),x)-(x-2*v-1)*diff(y(x),x)+(x-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{x} c_{1}+c_{2} x^{v +1} \BesselI \left (-v -1, x\right )+c_{3} x^{v +1} \BesselK \left (v +1, x\right ) \]
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 83
DSolve[(-1 + x)*y[x] - (-1 - 2*v + x)*y'[x] - (2*v + x)*y''[x] + x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 2^{-2 (v+1)} e^x G_{2,3}^{2,1}\left (2 x\left | {c} 1,v+\frac {3}{2} \\ 1,2 (v+1),0 \\ \\ \right .\right )+\frac {\sqrt {\pi } c_3 2^{-v-1} x^{v+1} I_{v+1}(x)}{\Gamma \left (\frac {1}{2}-v\right )}+c_1 e^x \\ \end{align*}