64.13.27 problem 27

Internal problem ID [13478]
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 : Wednesday, March 05, 2025 at 10:02:18 PM
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: 0.026 (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,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (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]
 

\[ y(x)\to \frac {2 x^5+x^2-6 x^2 \log (x)+3}{36 x^2} \]

✓ Sympy. Time used: 0.217 (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)
 

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