60.4.3 problem 1451

Internal problem ID [11454]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1451
Date solved : Tuesday, January 28, 2025 at 06:06:31 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-a \,x^{b} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.250 (sec). Leaf size: 114

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

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

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 164

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

\[ y(x)\to (-1)^{\frac {1}{b+3}} (b+3)^{-\frac {6}{b+3}} x a^{\frac {1}{b+3}} \left ((-1)^{\frac {1}{b+3}} c_3 x a^{\frac {1}{b+3}} \, _0F_2\left (;1+\frac {1}{b+3},1+\frac {2}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )+(b+3)^{\frac {3}{b+3}} c_2 \, _0F_2\left (;1-\frac {1}{b+3},1+\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )\right )+c_1 \, _0F_2\left (;1-\frac {2}{b+3},1-\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right ) \]