Internal problem ID [7166]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 29.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-y x=x^{6}-64} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 149
dsolve(diff(y(x),x$2)-x*y(x)-x^6+64=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {16 x^{7} \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {7}{3}\right ], \left [\frac {2}{3}, \frac {10}{3}\right ], \frac {x^{3}}{9}\right )-21 x^{8} \Gamma \left (\frac {2}{3}\right )^{2} \left (3^{\frac {1}{6}} \operatorname {AiryBi}\left (x \right )+3^{\frac {2}{3}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {8}{3}\right ], \left [\frac {4}{3}, \frac {11}{3}\right ], \frac {x^{3}}{9}\right )-7168 x \pi \left (\operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}}-3^{\frac {5}{6}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+5376 \Gamma \left (\frac {2}{3}\right ) \left (x^{2} \Gamma \left (\frac {2}{3}\right ) \left (3^{\frac {1}{6}} \operatorname {AiryBi}\left (x \right )+3^{\frac {2}{3}} \operatorname {AiryAi}\left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )+\frac {\operatorname {AiryBi}\left (x \right ) c_{1}}{16}+\frac {\operatorname {AiryAi}\left (x \right ) c_{2}}{16}\right )}{336 \Gamma \left (\frac {2}{3}\right )} \]
✓ Solution by Mathematica
Time used: 0.493 (sec). Leaf size: 256
DSolve[y''[x]-x*y[x]-x^6+64==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {192 \sqrt [3]{3} \pi x \operatorname {Gamma}\left (\frac {1}{3}\right ) \left (\sqrt {3} \operatorname {AiryAi}(x)-\operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{9}\right )-\frac {\sqrt [6]{3} \pi x^8 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {8}{3}\right ) \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {8}{3};\frac {4}{3},\frac {11}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {11}{3}\right )}+\frac {3 \sqrt [3]{3} \pi x^7 \operatorname {Gamma}\left (\frac {4}{3}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (\operatorname {AiryBi}(x)-\sqrt {3} \operatorname {AiryAi}(x)\right ) \, _1F_2\left (\frac {7}{3};\frac {2}{3},\frac {10}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {10}{3}\right )}+\frac {64 \sqrt [6]{3} \pi x^2 \operatorname {Gamma}\left (\frac {2}{3}\right )^2 \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {5}{3}\right )}+27 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right ) (c_1 \operatorname {AiryAi}(x)+c_2 \operatorname {AiryBi}(x))}{27 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right )} \]