Internal problem ID [10895]
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: 71.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(x*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*((a-c)*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-c x}+c_{2} x^{-\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {b}{2}, \frac {1}{2}-\frac {b}{2}, \left (-2 c +a \right ) x \right ) {\mathrm e}^{-\frac {a x}{2}} \]
✓ Solution by Mathematica
Time used: 0.333 (sec). Leaf size: 50
DSolve[x*y''[x]+(a*x+b)*y'[x]+c*((a-c)*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-c x} \left (c_1-c_2 x^{1-b} (x (a-2 c))^{b-1} \Gamma (1-b,(a-2 c) x)\right ) \]