Internal problem ID [9835]
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]]
\[ \boxed {x^{2} y^{\prime \prime \prime }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.062 (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)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{\frac {x}{2}} \sqrt {x}\, \operatorname {BesselI}\left (\nu , \frac {x}{2}\right )+c_{3} {\mathrm e}^{\frac {x}{2}} \sqrt {x}\, \operatorname {BesselK}\left (\nu , \frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.117 (sec). Leaf size: 80
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]
\[ y(x)\to e^x \left (\frac {c_3 x^{\nu +\frac {1}{2}} \operatorname {Gamma}\left (\nu +\frac {1}{2}\right ) \, _1\tilde {F}_1\left (\nu +\frac {1}{2};2 \nu +1;-x\right )}{\operatorname {Gamma}\left (\frac {3}{2}-\nu \right )}+c_2 G_{2,3}^{2,1}\left (x\left | \begin {array}{c} 1,0 \\ \frac {1}{2}-\nu ,\nu +\frac {1}{2},0 \\ \end {array} \right .\right )+c_1\right ) \]