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 = 0.221714 (sec), leaf count = 46
\[\left \{\left \{y(x)\to e^{-a x} (c_1 U(m,m+n,(a-b) x)+c_2 L_{-m}^{m+n-1}((a-b) x))\right \}\right \}\]
Maple ✓
cpu = 0.439 (sec), leaf count = 43
\[[y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-a x} \KummerM \left (m , n +m , x \left (a -b \right )\right )+\textit {\_C2} \,{\mathrm e}^{-a x} \KummerU \left (m , n +m , x \left (a -b \right )\right )]\] 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))
Maple raw output
[y(x) = _C1*exp(-a*x)*KummerM(m,n+m,x*(a-b))+_C2*exp(-a*x)*KummerU(m,n+m,x*(a-b))]