Internal problem ID [8705]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1127.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime } x +\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 31
dsolve(x*diff(diff(y(x),x),x)+(2*a*x*ln(x)+1)*diff(y(x),x)+(a^2*x*ln(x)^2+a*ln(x)+a)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-a x} {\mathrm e}^{a x}+c_{2} x^{-a x} {\mathrm e}^{a x} \ln \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 25
DSolve[(a + a*Log[x] + a^2*x*Log[x]^2)*y[x] + (1 + 2*a*x*Log[x])*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{a x} x^{-a x} (c_2 \log (x)+c_1) \\ \end{align*}