15.22.12 problem 12

Internal problem ID [3346]
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 : 12
Date solved : Sunday, March 30, 2025 at 01:37:33 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+2 y y^{\prime }&=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: 14

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

\[ y = x -\frac {1}{3} x^{3}+\frac {2}{15} x^{5}+\operatorname {O}\left (x^{7}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 19

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

\[ y(x)\to \frac {2 x^5}{15}-\frac {x^3}{3}+x \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x)*Derivative(y(x), x) + 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 2*y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular