67.24.18 problem 34.7 (d)

Internal problem ID [16985]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.7 (d)
Date solved : Thursday, October 02, 2025 at 01:41:31 PM
CAS classification : [_Titchmarsh]

\begin{align*} y^{\prime \prime }+y \ln \left (x \right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 55

Order:=6; 
ode:=diff(diff(y(x),x),x)+ln(x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1-\frac {\left (-1+x \right )^{3}}{6}+\frac {\left (-1+x \right )^{4}}{24}-\frac {\left (-1+x \right )^{5}}{60}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{4}}{12}+\frac {\left (-1+x \right )^{5}}{40}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 60

ode=D[y[x],{x,2}]+Log[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (-\frac {1}{60} (x-1)^5+\frac {1}{24} (x-1)^4-\frac {1}{6} (x-1)^3+1\right )+c_2 \left (\frac {1}{40} (x-1)^5-\frac {1}{12} (x-1)^4+x-1\right ) \]

✓ Sympy. Time used: 0.198 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*log(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\left (x - 1\right )^{4} \log {\left (x + 1 \right )}^{2}}{24} - \frac {\left (x - 1\right )^{2} \log {\left (x + 1 \right )}}{2} + 1\right ) + C_{1} \left (x - \frac {\left (x - 1\right )^{3} \log {\left (x + 1 \right )}}{6} - 1\right ) + O\left (x^{6}\right ) \]