82.2.21 problem 24-22

Internal problem ID [21782]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 24. Power series about an ordinary point. Page 719
Problem number : 24-22
Date solved : Thursday, October 02, 2025 at 08:02:06 PM
CAS classification : [NONE]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 33

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2*exp(x) + Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, 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 -y(x)**2*exp(x) + Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular