Internal problem ID [9062]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1483.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 37
dsolve(2*x*diff(diff(diff(y(x),x),x),x)-4*(x+nu-1)*diff(diff(y(x),x),x)+(2*x+6*nu-5)*diff(y(x),x)+(1-2*nu)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{\frac {x}{2}} x^{\nu } \BesselI \left (\nu , \frac {x}{2}\right )+c_{3} {\mathrm e}^{\frac {x}{2}} x^{\nu } \BesselK \left (\nu , \frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.083 (sec). Leaf size: 90
DSolve[(1 - 2*nu)*y[x] + (-5 + 6*nu + 2*x)*y'[x] - 4*(-1 + nu + x)*y''[x] + 2*x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} e^x \left (3 c_2 G_{2,3}^{2,1}\left (x\left | {c} 1,3 \nu -\frac {1}{2} \\ 1,2 \nu ,0 \\ \\ \right .\right )-\frac {2 c_3 \Gamma \left (\frac {5}{2}-3 \nu \right ) \left (\, _1F_1\left (\frac {3}{2}-3 \nu ;1-2 \nu ;-x\right )-1\right )}{\Gamma (2-2 \nu ) \Gamma \left (\frac {3}{2}-\nu \right )}+3 c_1\right ) \\ \end{align*}