48.3.4 problem Example 3.33

Internal problem ID [7528]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page 181
Problem number : Example 3.33
Date solved : Sunday, March 30, 2025 at 12:13:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 18

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

\begin{align*} y &= 0 \\ y &= x c_2 \,{\mathrm e}^{-c_1 x +1} \\ \end{align*}

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 15

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

\[ y(x)\to c_2 x e^{c_1 x} \]

✗ Sympy

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

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