Internal problem ID [10898]
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: 64.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 73
dsolve(x*diff(y(x),x$2)+a*diff(y(x),x)+(b*x+c)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-i \sqrt {b}\, x} \operatorname {KummerM}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right )+c_{2} {\mathrm e}^{-i \sqrt {b}\, x} \operatorname {KummerU}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.099 (sec). Leaf size: 85
DSolve[x*y''[x]+a*y'[x]+(b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-i \sqrt {b} x} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (a+\frac {i c}{\sqrt {b}}\right ),a,2 i \sqrt {b} x\right )+c_2 L_{-\frac {a}{2}-\frac {i c}{2 \sqrt {b}}}^{a-1}\left (2 i \sqrt {b} x\right )\right ) \]