47.5.7 problem 7

Internal problem ID [7496]
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
Date solved : Tuesday, January 28, 2025 at 03:10:49 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y&=x +\frac {1}{x} \end{align*}

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 640

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 = \frac {-5 \left (\sqrt {2}-\frac {6}{5}\right ) \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) x^{-\sqrt {2}} \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 )+\operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) x \left (\sqrt {2}-1\right ) \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right )\right )}{\left (-7 \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \sqrt {2}+4 \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) x \left (\sqrt {2}-\frac {11}{8}\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 ) x \left (\sqrt {2}+\frac {11}{8}\right ) \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right )}d x \right )+5 \left (\sqrt {2}+\frac {6}{5}\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 {x^{-1-\sqrt {2}} \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 [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+x \left (1+\sqrt {2}\right ) \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right )\right )}{\left (-7 \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \sqrt {2}+4 \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) x \left (\sqrt {2}-\frac {11}{8}\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 ) x \left (\sqrt {2}+\frac {11}{8}\right ) \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right )}d x \right )+2 x^{-\sqrt {2}} \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) c_{2} +2 x^{\sqrt {2}} \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) c_{1}}{2 x} \]

✓ Solution by Mathematica

Time used: 7.806 (sec). Leaf size: 636

DSolve[x^2*(x+1)*D[y[x],{x,2}]+x*(4*x+3)*D[y[x],x]-y[x]==x+1/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^{-1-\sqrt {2}} \left (x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right ) \int _1^x\frac {7 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) K[2]^{-1-\sqrt {2}} \left (K[2]^2+1\right )}{(K[2]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[2]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right ) K[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[2]\right ) K[2]\right )\right )}dK[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right ) \int _1^x-\frac {7 \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]^{-1+\sqrt {2}} \left (K[1]^2+1\right )}{(K[1]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[1]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[1]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[1]\right ) K[1]\right )\right )}dK[1]+c_2 x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right )+c_1 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right )\right ) \]