56.4.20 problem 20

Internal problem ID [8909]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 20
Date solved : Sunday, March 30, 2025 at 01:53:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y&=1 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-x*y(x) = 1; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 360

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

\[ y(x)\to c_2 \left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_1 \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (-\frac {137 x^6}{1990656000}+\frac {x^5}{4608000}+\frac {x^4}{73728}+\frac {x^3}{1728}+\frac {x^2}{64}+\frac {x}{4}+\frac {\log (x)}{2}\right ) \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \left (\frac {137 x^6 (6 \log (x)+5)}{11943936000}+\frac {x^5 (113-30 \log (x))}{138240000}+\frac {x^4 (41-12 \log (x))}{884736}+\frac {x^3 (3-\log (x))}{1728}+\frac {1}{128} x^2 (5-2 \log (x))+\frac {1}{4} x (2-\log (x))-\frac {1}{4} \log (x) (\log (x)+2)\right ) \]

✗ Sympy

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

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