Internal problem ID [10907]
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: 73.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 43
dsolve(x*diff(y(x),x$2)+((a+b)*x+n+m)*diff(y(x),x)+(a*b*x+a*n+b*m)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-a x} \operatorname {KummerM}\left (m , n +m , \left (a -b \right ) x \right )+c_{2} {\mathrm e}^{-a x} \operatorname {KummerU}\left (m , n +m , \left (a -b \right ) x \right ) \]
✓ Solution by Mathematica
Time used: 0.131 (sec). Leaf size: 46
DSolve[x*y''[x]+((a+b)*x+n+m)*y'[x]+(a*b*x+a*n+b*m)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} (c_1 \operatorname {HypergeometricU}(m,m+n,(a-b) x)+c_2 L_{-m}^{m+n-1}((a-b) x)) \]