60.4.82 problem 1532

Internal problem ID [11533]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1532
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: 0.010 (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: 0.008 (sec). Leaf size: 103

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

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