61.28.47 problem 107

Internal problem ID [12607]
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 : 107
Date solved : Tuesday, January 28, 2025 at 08:02:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.108 (sec). Leaf size: 166

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

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

✓ Solution by Mathematica

Time used: 0.830 (sec). Leaf size: 58

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

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