Internal problem ID [8975]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1396.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {A y}{\left (a \,x^{2}+x b +c \right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 189
dsolve(diff(diff(y(x),x),x) = -A/(a*x^2+b*x+c)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {a \,x^{2}+b x +c}\, \left (\frac {i \sqrt {4 c a -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 c a -b^{2}}}\right )^{\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}+c_{2} \sqrt {a \,x^{2}+b x +c}\, \left (\frac {i \sqrt {4 c a -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 c a -b^{2}}}\right )^{-\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \]
✓ Solution by Mathematica
Time used: 0.662 (sec). Leaf size: 199
DSolve[y''[x] == -((A*y[x])/(c + b*x + a*x^2)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {x (a x+b)+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \text {ArcTan}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right ) \left (c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \text {ArcTan}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\frac {c_2}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}\right ) \\ \end{align*}