74.16.20 problem 22

Internal problem ID [16455]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 22
Date solved : Monday, March 31, 2025 at 02:53:46 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\left (\frac {{y^{\prime }}^{2}}{3}-1\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 26

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

\[ y(x)\to \frac {x^5}{40}-\frac {x^3}{6}-\frac {x^2}{2}+1 \]

✗ Sympy

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

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