Internal problem ID [9783]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1456.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 73
dsolve(diff(diff(diff(y(x),x),x),x)+x^(2*c-2)*diff(y(x),x)+(c-1)*x^(2*c-3)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}+c_{2} \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}+c_{3} \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right ) \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 183
DSolve[(-1 + c)*x^(-3 + 2*c)*y[x] + x^(-2 + 2*c)*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \, _1F_2\left (\frac {1}{2}-\frac {1}{2 c};1-\frac {1}{c},1-\frac {1}{2 c};-\frac {x^{2 c}}{4 c^2}\right )+4^{-1/c} c^{-2/c} c_3 \left (x^{2 c}\right )^{\frac {1}{c}} \, _1F_2\left (\frac {1}{2}+\frac {1}{2 c};1+\frac {1}{2 c},1+\frac {1}{c};-\frac {x^{2 c}}{4 c^2}\right )+2^{-1/c} c^{-1/c} c_2 \left (x^{2 c}\right )^{\left .\frac {1}{2}\right /c} \, _1F_2\left (\frac {1}{2};1-\frac {1}{2 c},1+\frac {1}{2 c};-\frac {x^{2 c}}{4 c^2}\right ) \]