15.18.33 problem 33

Internal problem ID [3276]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 33
Date solved : Sunday, March 30, 2025 at 01:26:14 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

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

✓ Maple. Time used: 0.224 (sec). Leaf size: 14

ode:=y(x)*diff(diff(y(x),x),x)-y(x)^2*diff(y(x),x) = diff(y(x),x)^2; 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\frac {6}{{\mathrm e}^{\frac {3 x}{2}}-4} \]

✓ Mathematica. Time used: 1.99 (sec). Leaf size: 18

ode=y[x]*D[y[x],{x,2}]-y[x]^2*D[y[x],x]==D[y[x],x]^2; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : The given ODE -sqrt((y(x)**3 + 4*Derivative(y(x), (x, 2)))*y(x))/2 + y(x)**2/2 + Derivative(y(x), x) cannot be solved by the factorable group method