Internal problem ID [9080]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1501.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 37
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-2*(x^2-x)*diff(diff(y(x),x),x)+(x^2-2*x+1/4-nu^2)*diff(y(x),x)+(nu^2-1/4)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{x} c_{1}+c_{2} {\mathrm e}^{\frac {x}{2}} \sqrt {x}\, \BesselI \left (\nu , \frac {x}{2}\right )+c_{3} {\mathrm e}^{\frac {x}{2}} \sqrt {x}\, \BesselK \left (\nu , \frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.116 (sec). Leaf size: 76
DSolve[(-1/4 + nu^2)*y[x] + (1/4 - nu^2 - 2*x + x^2)*y'[x] - 2*(-x + x^2)*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^x \left (c_2 G_{2,3}^{2,1}\left (x\left | {c} 1,0 \\ \frac {1}{2}-\nu ,\nu +\frac {1}{2},0 \\ \\ \right .\right )+c_1\right )+\frac {\sqrt {\pi } c_3 e^{x/2} \sqrt {x} I_{\nu }\left (\frac {x}{2}\right )}{\Gamma \left (\frac {3}{2}-\nu \right )} \\ \end{align*}