82.1.15 problem 23-18

Internal problem ID [21757]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 23. Power series. Page 695
Problem number : 23-18
Date solved : Thursday, October 02, 2025 at 08:01:52 PM
CAS classification : [NONE]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

Order:=6; 
ode:=diff(diff(y(x),x),x) = x*y(x)^2-diff(y(x),x); 
ic:=[y(0) = 2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 2+x -\frac {1}{2} x^{2}+\frac {5}{6} x^{3}+\frac {1}{8} x^{4}-\frac {3}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {3 x^5}{40}+\frac {x^4}{8}+\frac {5 x^3}{6}-\frac {x^2}{2}+x+2 \]

✗ Sympy

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

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