61.32.25 problem 234

Internal problem ID [12735]
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
Problem number : 234
Date solved : Tuesday, January 28, 2025 at 04:17:10 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+A y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 174

dsolve((a*x^2+b*x+c)^2*diff(y(x),x$2)+A*y(x)=0,y(x), singsol=all)
 

\[ y = \left ({\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{2} +{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} \right ) \sqrt {a \,x^{2}+b x +c} \]

✓ Solution by Mathematica

Time used: 0.421 (sec). Leaf size: 154

DSolve[(a*x^2+b*x+c)^2*D[y[x],{x,2}]+A*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right ) \]