9.25.7 problem 6

Internal problem ID [3394]
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 : 6
Date solved : Tuesday, September 30, 2025 at 06:37:32 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }+y&=-2 x^{2}+x \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 64

Order:=6; 
ode:=x^2*(1+x)*diff(diff(y(x),x),x)+x*(x^2+3)*diff(y(x),x)+y(x) = -2*x^2+x; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {x^{2} \left (\frac {1}{4}-\frac {2}{9} x +\frac {7}{576} x^{2}+\frac {107}{7200} x^{3}-\frac {1031}{172800} x^{4}+\operatorname {O}\left (x^{5}\right )\right )+\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {1}{4} x^{2}-\frac {1}{64} x^{4}+\frac {3}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (7 x -x^{2}+\frac {7}{36} x^{3}+\frac {35}{1152} x^{4}-\frac {191}{9000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 300

ode=x^2*(x+1)*D[y[x],{x,2}]+x*(x^2+3)*D[y[x],x]+y[x]==x-2*x^2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (\frac {3 x^5}{800}-\frac {x^4}{64}+\frac {x^2}{4}-2 x+1\right )}{x}+\frac {c_1 \left (x^5 \left (\frac {3 \log (x)}{800}-\frac {629}{36000}\right )+x^4 \left (\frac {17}{1152}-\frac {\log (x)}{64}\right )+\frac {7 x^3}{36}+x^2 \left (\frac {\log (x)}{4}-\frac {3}{4}\right )+x (5-2 \log (x))+\log (x)+1\right )}{x}+\frac {\left (\frac {34109 x^6}{1152}-\frac {491 x^5}{30}+\frac {123 x^4}{16}-\frac {8 x^3}{3}+\frac {x^2}{2}\right ) \left (x^5 \left (\frac {3 \log (x)}{800}-\frac {629}{36000}\right )+x^4 \left (\frac {17}{1152}-\frac {\log (x)}{64}\right )+\frac {7 x^3}{36}+x^2 \left (\frac {\log (x)}{4}-\frac {3}{4}\right )+x (5-2 \log (x))+\log (x)+1\right )}{x}+\frac {\left (\frac {3 x^5}{800}-\frac {x^4}{64}+\frac {x^2}{4}-2 x+1\right ) \left (\frac {x^6 (33586-34109 \log (x))}{1152}+\frac {1}{900} x^5 (14730 \log (x)-12641)+\frac {1}{64} x^4 (319-492 \log (x))+\frac {1}{9} x^3 (24 \log (x)-5)+\frac {1}{4} x^2 (-2 \log (x)-1)\right )}{x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + 2*x**2 + x*(x**2 + 3)*Derivative(y(x), x) - x + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**2*(x + 1)*Derivative(y(x), (x, 2)) + 2*x**2 + x*(x**2 + 3)*Derivative(y(x), x) - x + y(x) does not match hint 2nd_power_series_regular