15.25.5 problem 4

Internal problem ID [3392]
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 : 4
Date solved : Sunday, March 30, 2025 at 01:38:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-2 x \right ) y^{\prime \prime }+4 x y^{\prime }-4 y&=x^{2}-x \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 53

Order:=6; 
ode:=(1-2*x)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-4*y(x) = x^2-x; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}\right ) y \left (0\right )+x y^{\prime }\left (0\right )-\frac {x^{3}}{6}-\frac {x^{4}}{12}-\frac {x^{5}}{30}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 60

ode=(1-2*x)*D[y[x],{x,2}]+4*x*D[y[x],x]-4*y[x]==x^2-x; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {x^5}{30}-\frac {x^4}{12}-\frac {x^3}{6}+c_1 \left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+1\right )+c_2 x \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + 4*x*Derivative(y(x), x) + x + (1 - 2*x)*Derivative(y(x), (x, 2)) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -x**2 + 4*x*Derivative(y(x), x) + x + (1 - 2*x)*Derivative(y(x), (x, 2)) - 4*y(x) does not match hint 2nd_power_series_regular