87.18.28 problem 28

Internal problem ID [23669]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 135
Problem number : 28
Date solved : Thursday, October 02, 2025 at 09:44:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x \left (1-2 x \ln \left (x \right )\right ) y^{\prime \prime }+\left (1+4 x^{2} \ln \left (x \right )\right ) y^{\prime }-\left (4 x +2\right ) y&={\mathrm e}^{2 x} \left (1-2 x \ln \left (x \right )\right )^{2} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\ln \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (\frac {1}{2}\right )&=\frac {{\mathrm e}}{2} \\ y^{\prime }\left (\frac {1}{2}\right )&={\mathrm e} \left (2+\ln \left (2\right )\right ) \\ \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 22

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

\[ y = \frac {\left (\left (-2 x +1\right ) \ln \left (x \right )+2 x \right ) {\mathrm e}^{2 x}}{2} \]

✗ Mathematica

ode=x*(1-2*x*Log[x])*D[y[x],{x,2}]+(1+4*x^2*Log[x])*D[y[x],x]-(2+4*x)*y[x]==Exp[2*x]*(1-2*x*Log[x])^2; 
ic={y[1/2]==Exp[1],Derivative[1][y][1/2] ==Exp[1]*(2+Log[2])}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(-2*x*log(x) + 1)*Derivative(y(x), (x, 2)) - (4*x + 2)*y(x) - (-2*x*log(x) + 1)**2*exp(2*x) + (4*x**2*log(x) + 1)*Derivative(y(x), x),0) 
ics = {y(1/2): E, Subs(Derivative(y(x), x), x, 1/2): E*(log(2) + 2)} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (4*x**2*exp(2*x)*log(x)**2 + 2*x**2*log(x)