Internal problem ID [5828]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3+4 x \right ) y^{\prime }-y=x +\frac {1}{x}} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 644
dsolve(x^2*(x+1)*diff(y(x),x$2)+x*(4*x+3)*diff(y(x),x)-y(x)=x+1/x,y(x), singsol=all)
\[ y \left (x \right ) = x^{-1-\sqrt {2}} \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) c_{2} +x^{\sqrt {2}-1} \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) c_{1} +\frac {3 \sqrt {2}\, \left (x^{-\sqrt {2}} \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}-\frac {5}{3}\right ) \left (\int \frac {x^{\sqrt {2}-1} \left (x^{2}+1\right ) \left (\left (\left (-2 x^{2}-4 x -\frac {5}{2}\right ) \sqrt {2}-x^{2}-\frac {11 x}{2}-3\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+\left (\sqrt {2}-1\right ) \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right ) x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right )\right )}{\left (-7 \sqrt {2}\, \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+4 \left (\sqrt {2}-\frac {11}{8}\right ) x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right )\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+4 \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}+\frac {11}{8}\right ) x}d x \right )+x^{\sqrt {2}} \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\int \frac {\left (\left (\left (-2 x^{2}-4 x -\frac {5}{2}\right ) \sqrt {2}+x^{2}+\frac {11 x}{2}+3\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+\operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \left (1+\sqrt {2}\right ) \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right ) x \right ) x^{-1-\sqrt {2}} \left (x^{2}+1\right )}{\left (-7 \sqrt {2}\, \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+4 \left (\sqrt {2}-\frac {11}{8}\right ) x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right )\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+4 \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}+\frac {11}{8}\right ) x}d x \right ) \left (\sqrt {2}+\frac {5}{3}\right )\right )}{2 x} \]
✓ Solution by Mathematica
Time used: 7.882 (sec). Leaf size: 636
DSolve[x^2*(x+1)*y''[x]+x*(4*x+3)*y'[x]-y[x]==x+1/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{-1-\sqrt {2}} \left (x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right ) \int _1^x\frac {7 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) K[2]^{-1-\sqrt {2}} \left (K[2]^2+1\right )}{(K[2]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[2]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right ) K[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[2]\right ) K[2]\right )\right )}dK[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right ) \int _1^x-\frac {7 \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]^{-1+\sqrt {2}} \left (K[1]^2+1\right )}{(K[1]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[1]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[1]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[1]\right ) K[1]\right )\right )}dK[1]+c_2 x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right )+c_1 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right )\right ) \]