75.20.32 problem 671

Internal problem ID [17095]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 671
Date solved : Monday, March 31, 2025 at 03:41:26 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y&=\frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}} \end{align*}

With initial conditions

\begin{align*} y \left (\infty \right )&=0 \end{align*}

✓ Maple. Time used: 0.149 (sec). Leaf size: 21

ode:=2*x^2*(2-ln(x))*diff(diff(y(x),x),x)+x*(4-ln(x))*diff(y(x),x)-y(x) = (2-ln(x))^2/x^(1/2); 
ic:=y(infinity) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\ln \left (x \right ) c_2 \sqrt {x}-\ln \left (x \right )+1}{\sqrt {x}} \]

✗ Mathematica

ode=2*x^2*(2-Log[x])*D[y[x],{x,2}]+x*(4-Log[x])*D[y[x],x]-y[x]==(2-Log[x])^2/Sqrt[x]; 
ic={y[Infinity]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(2 - log(x))*Derivative(y(x), (x, 2)) + x*(4 - log(x))*Derivative(y(x), x) - y(x) - (2 - log(x))**2/sqrt(x),0) 
ics = {y(oo): 0} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*x**(5/2)*log(x)*Derivative(y(x), (x, 2)) + 4*x**(5/2)*Derivative(y(x), (x, 2)) - sqrt(x)*y(x) - log(x)**2 + 4*log(x) - 4)/(x**(3/2)*(log(x) - 4)) cannot be solved by the factorable group method