ODE
\[ y'(x) (x (a+b)+m+n)+y(x) (a b x+a n+b m)+x y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0913131 (sec), leaf count = 46
\[\left \{\left \{y(x)\to e^{-a x} \left (c_1 U(m,m+n,(a-b) x)+c_2 L_{-m}^{m+n-1}(x (a-b))\right )\right \}\right \}\]
Maple ✓
cpu = 0.124 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-ax}} \left ( {{\sl U}\left (m,\,n+m,\,x \left ( a-b \right ) \right )}{\it \_C2}+{{\sl M}\left (m,\,n+m,\,x \left ( a-b \right ) \right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(b*m + a*n + a*b*x)*y[x] + (m + n + (a + b)*x)*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*HypergeometricU[m, m + n, (a - b)*x] + C[2]*LaguerreL[-m, -1 + m + n, (a - b)*x])/E^(a*x)}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+(m+n+x*(a+b))*diff(y(x),x)+(a*b*x+a*n+b*m)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-a*x)*(KummerU(m,n+m,x*(a-b))*_C2+KummerM(m,n+m,x*(a-b))*_C1)