61.28.21 problem 81

Internal problem ID [12581]
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 : 81
Date solved : Tuesday, January 28, 2025 at 03:22:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 69

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

\[ y = \frac {{\mathrm e}^{\frac {b^{2}}{2 a}} \pi c_{2} \left (a x +b \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b \right )}{2 \sqrt {a}}\right )+\sqrt {a}\, \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} c_{2} +c_{1} \left (a x +b \right )}{x} \]

✓ Solution by Mathematica

Time used: 0.352 (sec). Leaf size: 71

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

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