Internal problem ID [8698]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1118.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x +\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
dsolve(x*diff(diff(y(x),x),x)+((a+b)*x+m+n)*diff(y(x),x)+(a*b*x+a*n+b*m)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-a x} \KummerM \left (m , m +n , x \left (a -b \right )\right )+c_{2} {\mathrm e}^{-a x} \KummerU \left (m , m +n , x \left (a -b \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 46
DSolve[(b*m + a*n + a*b*x)*y[x] + (m + n + (a + b)*x)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-a x} (c_1 \text {HypergeometricU}(m,m+n,x (a-b))+c_2 L_{-m}^{m+n-1}((a-b) x)) \\ \end{align*}