Internal problem ID [8965]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1386.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 68
dsolve(diff(diff(y(x),x),x) = 18/(2*x+1)^2/(x^2+x+1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (x^{2}+x +1\right )}{\left (2 x +1\right )^{2}}+\frac {c_{2} \left (\left (-36 x^{2}-36 x -36\right ) \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )+16 \sqrt {3}\, \left (x^{3}+x^{2}+\frac {11}{8} x +\frac {3}{16}\right )\right )}{\left (2 x +1\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 66
DSolve[y''[x] == (18*y[x])/((1 + 2*x)^2*(1 + x + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-12 \sqrt {3} c_2 \left (x^2+x+1\right ) \text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )+c_1 \left (x^2+x+1\right )+c_2 (2 x+1) (8 x (x+1)+11)}{(2 x+1)^2} \\ \end{align*}