15.25.3 problem 2

Internal problem ID [3390]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 2
Date solved : Sunday, March 30, 2025 at 01:38:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{3} \left (\frac {1}{9}+\frac {2}{225} x^{2}+\operatorname {O}\left (x^{3}\right )\right )+\left (-\frac {1}{2} x^{2}-\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 188

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

\[ y(x)\to c_2 \left (\frac {x^6}{288}+\frac {x^4}{16}+\frac {x^2}{2}+1\right )+c_1 \left (x^6 \left (\frac {\log (x)}{144}-\frac {1}{108}\right )+x^4 \left (\frac {\log (x)}{8}-\frac {1}{8}\right )+x^2 \left (\log (x)-\frac {1}{2}\right )+2 \log (x)+1\right )+\left (\frac {x^6}{288}+\frac {x^4}{16}+\frac {x^2}{2}+1\right ) \left (\frac {1}{100} x^5 (7-10 \log (x))+\frac {1}{18} x^3 (-6 \log (x)-1)\right )+\left (\frac {x^5}{20}+\frac {x^3}{6}\right ) \left (x^6 \left (\frac {\log (x)}{144}-\frac {1}{108}\right )+x^4 \left (\frac {\log (x)}{8}-\frac {1}{8}\right )+x^2 \left (\log (x)-\frac {1}{2}\right )+2 \log (x)+1\right ) \]

✗ Sympy

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

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