15.25.11 problem 10

Internal problem ID [3398]
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 : 10
Date solved : Sunday, March 30, 2025 at 01:39:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 57

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

\[ y = \frac {c_1 \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-1+\frac {1}{14} x^{3}+\frac {1}{126} x^{4}+\frac {1}{1386} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 248

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

\[ y(x)\to \frac {c_1 \left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}}+c_2 x \left (\frac {x^5}{45045}+\frac {x^4}{3465}+\frac {x^3}{315}+\frac {x^2}{35}+\frac {x}{5}+1\right )+\frac {\left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right ) \left (-\frac {6233 x^{11/2}}{1921920}+\frac {2107 x^{9/2}}{95040}-\frac {143 x^{7/2}}{1512}-\frac {x^{5/2}}{140}+\frac {x^{3/2}}{15}-\frac {2 \sqrt {x}}{3}\right )}{\sqrt {x}}+x \left (\frac {x^5}{45045}+\frac {x^4}{3465}+\frac {x^3}{315}+\frac {x^2}{35}+\frac {x}{5}+1\right ) \left (\frac {x^6}{691200}-\frac {2747 x^5}{518918400}+\frac {x^2}{6}-\frac {1}{3 x}\right ) \]

✗ Sympy

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

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