Internal problem ID [10941]
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 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 107.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a x +b \right ) y^{\prime \prime }+s \left (x c +d \right ) y^{\prime }-s^{2} \left (\left (c +a \right ) x +b +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 149
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)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {s \left (a +c \right ) x}{a}} \operatorname {KummerM}\left (1, \frac {-d s a +c s b +2 a^{2}}{a^{2}}, \frac {s \left (2 a +c \right ) \left (a x +b \right )}{a^{2}}\right ) \left (a x +b \right )^{\frac {-d s a +c s b +a^{2}}{a^{2}}}+c_{2} {\mathrm e}^{-\frac {s \left (a +c \right ) x}{a}} \operatorname {KummerU}\left (1, \frac {-d s a +c s b +2 a^{2}}{a^{2}}, \frac {s \left (2 a +c \right ) \left (a x +b \right )}{a^{2}}\right ) \left (a x +b \right )^{\frac {-d s a +c s b +a^{2}}{a^{2}}} \]
✓ Solution by Mathematica
Time used: 1.269 (sec). Leaf size: 122
DSolve[(a*x+b)*y''[x]+s*(c*x+d)*y'[x]-s^2*((a+c)*x+b+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{s x}-\frac {c_2 e^{s \left (\frac {b (2 a+c)}{a^2}+x\right )} (a x+b)^{\frac {s (b c-a d)}{a^2}+1} \left (\frac {s (2 a+c) (a x+b)}{a^2}\right )^{\frac {s (a d-b c)}{a^2}-1} \Gamma \left (\frac {a^2-d s a+b c s}{a^2},\frac {(2 a+c) s (b+a x)}{a^2}\right )}{a} \]