Internal problem ID [8734]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1156.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+\ln \left (x \right ) x \right )=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 73
dsolve(x^2*diff(diff(y(x),x),x)+y(x)/ln(x)-x*exp(x)*(2+x*ln(x))=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} \ln \left (x \right )+\left (-\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right ) \ln \left (x \right )-x \right ) c_{1} -\left (-\left (\int \frac {\left (\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right ) \ln \left (x \right )+x \right ) {\mathrm e}^{x} \left (2+\ln \left (x \right ) x \right )}{x}d x \right )+\ln \left (x \right ) {\mathrm e}^{x} \left (\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right ) \ln \left (x \right )+x \right )\right ) \ln \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 27
DSolve[-(E^x*x*(2 + x*Log[x])) + y[x]/Log[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \operatorname {LogIntegral}(x) \log (x)+c_2 (-x)+\left (e^x+c_1\right ) \log (x) \\ \end{align*}