60.4.31 problem 1479

Internal problem ID [11482]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1479
Date solved : Tuesday, January 28, 2025 at 06:06:35 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 92

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

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

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 153

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

\[ y(x)\to \frac {1}{2} i c_2 x \, _1F_2\left (\frac {a}{2}+\frac {1}{2};\frac {3}{2},\frac {a}{2}+\frac {b}{2}+\frac {1}{2};\frac {x^2}{4}\right )+c_1 \, _1F_2\left (\frac {a}{2};\frac {1}{2},\frac {a}{2}+\frac {b}{2};\frac {x^2}{4}\right )+c_3 \left (\frac {i}{2}\right )^{-a-b+2} x^{-a-b+2} \, _1F_2\left (1-\frac {b}{2};-\frac {a}{2}-\frac {b}{2}+\frac {3}{2},-\frac {a}{2}-\frac {b}{2}+2;\frac {x^2}{4}\right ) \]