61.30.30 problem 178

Internal problem ID [12678]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 178
Date solved : Tuesday, January 28, 2025 at 03:38:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.604 (sec). Leaf size: 88

dsolve((a*x^2+2*b*x+c)*diff(y(x),x$2)+3*(a*x+b)*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.110 (sec). Leaf size: 152

DSolve[(a*x^2+2*b*x+c)*D[y[x],{x,2}]+3*(a*x+b)*D[y[x],x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_1 P_{\frac {\sqrt {a-d}}{\sqrt {a}}-\frac {1}{2}}^{\frac {1}{2}}\left (\frac {\sqrt {-b^2-a c} (b+a x)}{a \sqrt {c^2-\frac {b^4}{a^2}}}\right )+c_2 Q_{\frac {\sqrt {a-d}}{\sqrt {a}}-\frac {1}{2}}^{\frac {1}{2}}\left (\frac {\sqrt {-b^2-a c} (b+a x)}{a \sqrt {c^2-\frac {b^4}{a^2}}}\right )}{\sqrt [4]{x (a x+2 b)+c}} \]