Internal problem ID [8697]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1117.
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 +b \right ) \left (x +1\right ) y^{\prime }+a b x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 93
dsolve(x*diff(diff(y(x),x),x)-(a+b)*(x+1)*diff(y(x),x)+a*b*x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{b x} \KummerM \left (\frac {a^{2}+\left (b +1\right ) a -b}{a -b}, b +2+a , x \left (a -b \right )\right ) x^{a +b +1}+c_{2} {\mathrm e}^{b x} \KummerU \left (\frac {a^{2}+\left (b +1\right ) a -b}{a -b}, b +2+a , x \left (a -b \right )\right ) x^{a +b +1} \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 77
DSolve[a*b*x*y[x] - (a + b)*(1 + x)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{b x} x^{a+b+1} \left (c_1 \text {HypergeometricU}\left (\frac {a (a+b)}{a-b}+1,a+b+2,x (a-b)\right )+c_2 L_{\frac {2 a^2}{b-a}+a-1}^{a+b+1}((a-b) x)\right ) \\ \end{align*}