Internal problem ID [9807]
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]]
\[ \boxed {x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (x -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 54
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)
\[ y \left (x \right ) = \frac {x^{2+v} c_{2} \operatorname {BesselI}\left (-v +1, x\right )-2 \operatorname {BesselI}\left (-v , x\right ) x^{v +1} c_{2} v +x^{2+v} c_{3} \operatorname {BesselK}\left (v +1, x\right )+{\mathrm e}^{x} c_{1} x}{x} \]
✓ Solution by Mathematica
Time used: 0.127 (sec). Leaf size: 91
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]
\[ y(x)\to \frac {1}{4} e^x \left (\frac {4 c_3 x^{2 v+2} \operatorname {Gamma}\left (v+\frac {3}{2}\right ) \, _1\tilde {F}_1\left (v+\frac {3}{2};2 v+3;-2 x\right )}{\operatorname {Gamma}\left (\frac {1}{2}-v\right )}+c_2 4^{-v} G_{2,3}^{2,1}\left (2 x\left | \begin {array}{c} 1,v+\frac {3}{2} \\ 1,2 (v+1),0 \\ \end {array} \right .\right )+4 c_1\right ) \]