Internal problem ID [8700]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1120.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (x c +d \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 123
dsolve(x*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+(c*x+d)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x \left (a +\sqrt {a^{2}-4 c}\right )}{2}} \KummerM \left (\frac {b \sqrt {a^{2}-4 c}+b a -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right )+c_{2} {\mathrm e}^{-\frac {x \left (a +\sqrt {a^{2}-4 c}\right )}{2}} \KummerU \left (\frac {b \sqrt {a^{2}-4 c}+b a -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 134
DSolve[(d + c*x)*y[x] + (b + a*x)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 c}+a\right )} \left (c_1 \text {HypergeometricU}\left (\frac {b \left (\sqrt {a^2-4 c}+a\right )-2 d}{2 \sqrt {a^2-4 c}},b,x \sqrt {a^2-4 c}\right )+c_2 L_{\frac {2 d-b \left (a+\sqrt {a^2-4 c}\right )}{2 \sqrt {a^2-4 c}}}^{b-1}\left (\sqrt {a^2-4 c} x\right )\right ) \\ \end{align*}