Internal problem ID [7184]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 47.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}-y x=x^{2}+\frac {1}{x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 28
dsolve(diff(y(x),x$2)-1/x*diff(y(x),x)-x*y(x)-x^2-1/x=0,y(x), singsol=all)
\[ y \left (x \right ) = x \operatorname {BesselI}\left (\frac {2}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right ) c_{2} +x \operatorname {BesselK}\left (\frac {2}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right ) c_{1} -x \]
✓ Solution by Mathematica
Time used: 0.487 (sec). Leaf size: 253
DSolve[y''[x]-1/x*y'[x]-x*y[x]-x^2-1/x==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\frac {3 \sqrt [6]{3} \pi \operatorname {Gamma}\left (-\frac {1}{3}\right ) \left (3 \operatorname {AiryAiPrime}(x)+\sqrt {3} \operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (-\frac {1}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{9}\right )}{x \operatorname {Gamma}\left (\frac {2}{3}\right )}+\frac {\frac {\sqrt [3]{3} \pi x \operatorname {Gamma}\left (\frac {1}{3}\right )^2 \left (\sqrt {3} \operatorname {AiryAiPrime}(x)-\operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {1}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {4}{3}\right )}+\frac {\sqrt [3]{3} \pi x^4 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right ) \left (\sqrt {3} \operatorname {AiryAiPrime}(x)-\operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {4}{3};\frac {5}{3},\frac {7}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {7}{3}\right )}+3 \sqrt [6]{3} \pi x^2 \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (3 \operatorname {AiryAiPrime}(x)+\sqrt {3} \operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {2}{3};\frac {1}{3},\frac {5}{3};\frac {x^3}{9}\right )+27 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) (c_1 \operatorname {AiryAiPrime}(x)+c_2 \operatorname {AiryBiPrime}(x))}{\operatorname {Gamma}\left (\frac {5}{3}\right )}}{27 \operatorname {Gamma}\left (\frac {1}{3}\right )} \]