61.28.25 problem 85

Internal problem ID [12585]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 85
Date solved : Tuesday, January 28, 2025 at 08:02:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.142 (sec). Leaf size: 143

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

\[ y = \frac {\left (\operatorname {hypergeom}\left (\left [\frac {3 a^{3}-a^{2} d +a b c -c^{2}}{2 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) \left (a^{2} x +a b -2 c \right ) c_{2} +c_{1} \operatorname {hypergeom}\left (\left [\frac {2 a^{3}-a^{2} d +a b c -c^{2}}{2 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right )\right ) {\mathrm e}^{-\frac {x \left (a^{2} x +2 a b -2 c \right )}{2 a}}}{x} \]

✓ Solution by Mathematica

Time used: 0.085 (sec). Leaf size: 134

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

\[ y(x)\to \frac {e^{-\frac {1}{2} x \left (-\frac {2 c}{a}+a x+2 b\right )} \left (c_2 \operatorname {Hypergeometric1F1}\left (-\frac {-2 a^3+d a^2-b c a+c^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+b a-2 c\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-2 a^3+d a^2-b c a+c^2}{a^3},\frac {x a^2+b a-2 c}{\sqrt {2} a^{3/2}}\right )\right )}{x} \]