77.1.116 problem 142 (page 205)

Internal problem ID [17935]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 142 (page 205)
Date solved : Monday, March 31, 2025 at 04:51:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 68

ode:=diff(diff(y(x),x),x)+1/x^2/ln(x)*y(x) = exp(x)*(2/x+ln(x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+1/(x^2*Log[x])*y[x]==Exp[x]*(2/x+Log[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \operatorname {LogIntegral}(x) \log (x)+c_2 (-x)+\left (e^x+c_1\right ) \log (x) \]

✗ Sympy

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

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