60.4.7 problem 1455

Internal problem ID [11458]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1455
Date solved : Tuesday, January 28, 2025 at 06:06:32 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.190 (sec). Leaf size: 71

dsolve(diff(diff(diff(y(x),x),x),x)-x^2*diff(diff(y(x),x),x)+(a+b-1)*x*diff(y(x),x)-b*y(x)*a=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 127

DSolve[-(a*b*y[x]) + (-1 + a + b)*x*D[y[x],x] - x^2*D[y[x],{x,2}] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt [3]{-\frac {1}{3}} c_2 x \, _2F_2\left (\frac {1}{3}-\frac {a}{3},\frac {1}{3}-\frac {b}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{3}\right )+c_1 \, _2F_2\left (-\frac {a}{3},-\frac {b}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{3}\right )+\left (-\frac {1}{3}\right )^{2/3} c_3 x^2 \, _2F_2\left (\frac {2}{3}-\frac {a}{3},\frac {2}{3}-\frac {b}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{3}\right ) \]