60.3.142 problem 1156

Internal problem ID [11138]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1156
Date solved : Sunday, March 30, 2025 at 07:43:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 68

ode:=x^2*diff(diff(y(x),x),x)+y(x)/ln(x)-x*exp(x)*(2+x*ln(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \int \frac {\left (\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right ) \ln \left (x \right )+x \right ) {\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )}{x}d x \ln \left (x \right )+\ln \left (x \right ) \left (-{\mathrm e}^{x} \ln \left (x \right )^{2}+c_1 \right ) \operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )-{\mathrm e}^{x} \ln \left (x \right )^{2} x +c_1 x +\ln \left (x \right ) c_2 \]

✓ Mathematica. Time used: 0.249 (sec). Leaf size: 76

ode=-(E^x*x*(2 + x*Log[x])) + y[x]/Log[x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \log (x) \left (\int _1^x-\frac {e^{K[2]} \log (K[2]) (K[2] \log (K[2])+2) \int _1^{K[2]}\frac {1}{\log ^2(K[1])}dK[1]}{K[2]}dK[2]+\left (e^x \log ^2(x)+c_2\right ) \int _1^x\frac {1}{\log ^2(K[1])}dK[1]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(x*log(x) + 2)*exp(x) + y(x)/log(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve x**2*Derivative(y(x), (x, 2)) - x*(x*log(x) + 2)*exp(x) + y(x)/log(x)