Internal problem ID [9447]
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]]
\[ \boxed {y^{\prime \prime } x -\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y=0} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 82
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)
\[ y \left (x \right ) = {\mathrm e}^{b x} x^{b +a +1} \left (\operatorname {KummerM}\left (\frac {a^{2}+a b +a -b}{a -b}, b +2+a , x \left (a -b \right )\right ) c_{1} +\operatorname {KummerU}\left (\frac {a^{2}+a b +a -b}{a -b}, b +2+a , x \left (a -b \right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 87
DSolve[a*b*x*y[x] - (a + b)*(1 + x)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{b x} x^{a+b+1} \left (c_1 \operatorname {HypergeometricU}\left (\frac {a^2+b a+a-b}{a-b},a+b+2,(a-b) x\right )+c_2 L_{-\frac {a^2+b a+a-b}{a-b}}^{a+b+1}((a-b) x)\right ) \]