56.5.12 problem 12

Internal problem ID [8973]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 12
Date solved : Sunday, March 30, 2025 at 01:57:38 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+y^{\prime }&=\frac {1}{x} \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 159

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

\[ y(x)\to -\frac {x^6}{4320}-\frac {x^5}{600}-\frac {x^4}{96}-\frac {x^3}{18}-\frac {x^2}{4}+c_2 \left (-\frac {x^5}{720}+\frac {x^4}{120}-\frac {x^3}{24}+\frac {x^2}{6}-\frac {x}{2}+1\right ) x+\left (-\frac {x^5}{720}+\frac {x^4}{120}-\frac {x^3}{24}+\frac {x^2}{6}-\frac {x}{2}+1\right ) x \left (\frac {x^6}{2160}+\frac {x^5}{600}+\frac {x^4}{96}+\frac {x^3}{18}+\frac {x^2}{4}+x+\log (x)\right )-x+c_1 \]

✗ Sympy

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

ValueError : ODE Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1/x does not match hint 2nd_power_series_regular