Internal problem ID [11069]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 234.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+A y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 189
dsolve((a*x^2+b*x+c)^2*diff(y(x),x$2)+A*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {a \,x^{2}+x b +c}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+c_{2} \sqrt {a \,x^{2}+x b +c}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} \]
✓ Solution by Mathematica
Time used: 2.154 (sec). Leaf size: 199
DSolve[(a*x^2+b*x+c)^2*y''[x]+A*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {x (a x+b)+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right ) \left (c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\frac {c_2}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}\right ) \]