Internal problem ID [8699]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1119.
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 -2 \left (a x +b \right ) y^{\prime }+\left (a^{2} x +2 a b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 24
dsolve(x*diff(diff(y(x),x),x)-2*(a*x+b)*diff(y(x),x)+(a^2*x+2*a*b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{a x}+c_{2} x^{2 b +1} {\mathrm e}^{a x} \]
✓ Solution by Mathematica
Time used: 0.072 (sec). Leaf size: 75
DSolve[(2*a*b + a^2*x)*y[x] - 2*(b + a*x)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{a x} x^{b-\frac {1}{2} \sqrt {(2 b+1)^2}+\frac {1}{2}} \left (c_2 x^{\sqrt {(2 b+1)^2}}+\sqrt {(2 b+1)^2} c_1\right )}{\sqrt {(2 b+1)^2}} \\ \end{align*}