Internal problem ID [8966]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1387.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {3 y}{4 \left (x^{2}+x +1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 36
dsolve(diff(diff(y(x),x),x) = 3/4/(x^2+x+1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x^{2}+x +1}+c_{2} \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {x^{2}+x +1} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 45
DSolve[y''[x] == (3*y[x])/(4*(1 + x + x^2)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \sqrt {x^2+x+1} \left (2 \sqrt {3} c_2 \text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )+3 c_1\right ) \\ \end{align*}