9.22.5 problem 5

Internal problem ID [3339]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 06:36:45 AM
CAS classification : [`y=_G(x,y')`]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 14

Order:=5; 
ode:=diff(y(x),x) = ln(x*y(x)); 
ic:=[y(1) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

\[ y = 1+\frac {1}{2} \left (x -1\right )^{2}+\frac {1}{12} \left (x -1\right )^{4}+\operatorname {O}\left (\left (x -1\right )^{5}\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 23

ode=D[y[x],x]==Log[x*y[x]]; 
ic={y[1]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,4}]
 

\[ y(x)\to \frac {1}{12} (x-1)^4+\frac {1}{2} (x-1)^2+1 \]

✓ Sympy. Time used: 0.191 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-log(x*y(x)) + Derivative(y(x), x),0) 
ics = {y(1): 1} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=1,n=5)
 

\[ y{\left (x \right )} = 1 + \frac {\left (x - 1\right )^{2}}{2} + \frac {\left (x - 1\right )^{4}}{12} + O\left (x^{5}\right ) \]