Internal problem ID [5075]
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]]
Solve \begin {gather*} \boxed {x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y-x -\frac {1}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 803
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)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 7.157 (sec). Leaf size: 568
DSolve[x^2*(x+1)*y''[x]+x*(4*x+3)*y'[x]-y[x]==x+1/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^{\sqrt {2}-1} \, _2F_1\left (-1+\sqrt {2},2+\sqrt {2};1+2 \sqrt {2};-x\right ) \left (\int _1^x\frac {7 \, _2F_1\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 ) \, _2F_1\left (-\sqrt {2},3-\sqrt {2};2-2 \sqrt {2};-K[2]\right ) \, _2F_1\left (-1+\sqrt {2},2+\sqrt {2};1+2 \sqrt {2};-K[2]\right ) K[2]+\, _2F_1\left (-1-\sqrt {2},2-\sqrt {2};1-2 \sqrt {2};-K[2]\right ) \left (14 \sqrt {2} \, _2F_1\left (-1+\sqrt {2},2+\sqrt {2};1+2 \sqrt {2};-K[2]\right )+\left (-4+\sqrt {2}\right ) \, _2F_1\left (\sqrt {2},3+\sqrt {2};2 \left (1+\sqrt {2}\right );-K[2]\right ) K[2]\right )\right )}dK[2]+c_2\right )+x^{-1-\sqrt {2}} \, _2F_1\left (-1-\sqrt {2},2-\sqrt {2};1-2 \sqrt {2};-x\right ) \left (\int _1^x-\frac {7 \, _2F_1\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 ) \, _2F_1\left (-\sqrt {2},3-\sqrt {2};2-2 \sqrt {2};-K[1]\right ) \, _2F_1\left (-1+\sqrt {2},2+\sqrt {2};1+2 \sqrt {2};-K[1]\right ) K[1]+\, _2F_1\left (-1-\sqrt {2},2-\sqrt {2};1-2 \sqrt {2};-K[1]\right ) \left (14 \sqrt {2} \, _2F_1\left (-1+\sqrt {2},2+\sqrt {2};1+2 \sqrt {2};-K[1]\right )+\left (-4+\sqrt {2}\right ) \, _2F_1\left (\sqrt {2},3+\sqrt {2};2 \left (1+\sqrt {2}\right );-K[1]\right ) K[1]\right )\right )}dK[1]+c_1\right ) \\ \end{align*}