15.22.14 problem 14

Internal problem ID [3348]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:37:36 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\frac {{y^{\prime }}^{2}}{2}-y&=0 \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: 0.003 (sec). Leaf size: 20

Order:=7; 
ode:=diff(diff(y(x),x),x)+1/2*diff(y(x),x)^2-y(x) = 0; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = x -\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{32} x^{4}+\frac {1}{20} x^{5}-\frac {13}{480} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 40

ode=D[y[x],{x,2}]+1/2*D[y[x],x]^2-y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,6}]
 

\[ y(x)\to -\frac {13 x^6}{480}+\frac {x^5}{20}-\frac {3 x^4}{32}+\frac {x^3}{4}-\frac {x^2}{4}+x \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + Derivative(y(x), x)**2/2 + 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=7)
 

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