35.8.24 problem 24

Internal problem ID [6231]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page 466
Problem number : 24
Date solved : Sunday, March 30, 2025 at 10:44:07 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 35

ode:=x*(y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2) = y(x)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \sqrt {c_1 \,x^{2}+2 c_2} \\ y &= -\sqrt {c_1 \,x^{2}+2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.245 (sec). Leaf size: 18

ode=x*(y[x]*D[y[x],{x,2}]+(D[y[x],x])^2)==y[x]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \sqrt {x^2+c_1} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2) - y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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