58.13.27 problem 27

Internal problem ID [14838]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.5. The Cauchy-Euler Equation. Exercises page 169
Problem number : 27
Date solved : Thursday, October 02, 2025 at 09:55:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-6 y&=\ln \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&={\frac {1}{6}} \\ y^{\prime }\left (1\right )&=-{\frac {1}{6}} \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 20

ode:=x^2*diff(diff(y(x),x),x)-6*y(x) = ln(x); 
ic:=[y(1) = 1/6, D(y)(1) = -1/6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{3}}{18}+\frac {1}{12 x^{2}}-\frac {\ln \left (x \right )}{6}+\frac {1}{36} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 29

ode=x^2*D[y[x],{x,2}]-6*y[x]==Log[x]; 
ic={y[1]==1/6,Derivative[1][y][1]==-1/6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 x^5+x^2-6 x^2 \log (x)+3}{36 x^2} \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 6*y(x) - log(x),0) 
ics = {y(1): 1/6, Subs(Derivative(y(x), x), x, 1): -1/6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{3}}{18} - \frac {\log {\left (x \right )}}{6} + \frac {1}{36} + \frac {1}{12 x^{2}} \]