60.4.83 problem 1533

Internal problem ID [11534]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1533
Date solved : Tuesday, January 28, 2025 at 06:06:47 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-x y^{\prime }-n y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 58

dsolve(diff(diff(diff(y(x),x),x),x)-x*diff(y(x),x)-n*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{9}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\frac {n}{3}+\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+c_3 \,x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 106

DSolve[-(n*y[x]) - x*D[y[x],x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{9} \left (3 \sqrt [3]{-3} c_2 x \, _1F_2\left (\frac {n}{3}+\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{9}\right )+9 c_1 \, _1F_2\left (\frac {n}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{9}\right )+(-3)^{2/3} c_3 x^2 \, _1F_2\left (\frac {n}{3}+\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )\right ) \]