Internal problem ID [10908]
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: 74.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (x c +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 123
dsolve(x*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+(c*x+d)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {x \left (a +\sqrt {a^{2}-4 c}\right )}{2}} \operatorname {KummerM}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right )+c_{2} {\mathrm e}^{-\frac {x \left (a +\sqrt {a^{2}-4 c}\right )}{2}} \operatorname {KummerU}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.135 (sec). Leaf size: 135
DSolve[x*y''[x]+(a*x+b)*y'[x]+(c*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 c}+a\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {a b+\sqrt {a^2-4 c} b-2 d}{2 \sqrt {a^2-4 c}},b,\sqrt {a^2-4 c} x\right )+c_2 L_{-\frac {a b+\sqrt {a^2-4 c} b-2 d}{2 \sqrt {a^2-4 c}}}^{b-1}\left (\sqrt {a^2-4 c} x\right )\right ) \]