82.40.7 problem Ex. 7

Internal problem ID [18874]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 94
Problem number : Ex. 7
Date solved : Thursday, March 13, 2025 at 01:05:06 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2}-3 y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 55

ode:=x^2*y(x)*diff(diff(y(x),x),x)+(-y(x)+x*diff(y(x),x))^2-3*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {10}\, \sqrt {-x \left (c_{1} x^{5}-c_{2} \right )}}{5 x} \\ y \left (x \right ) &= \frac {\sqrt {10}\, \sqrt {-x \left (c_{1} x^{5}-c_{2} \right )}}{5 x} \\ \end{align*}

✓ Mathematica. Time used: 0.302 (sec). Leaf size: 23

ode=x^2*y[x]*D[y[x],{x,2}]+(x*D[y[x],x]-y[x])^2-3*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_2 \sqrt {x^5+c_1}}{\sqrt {x}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x)*Derivative(y(x), (x, 2)) + (x*Derivative(y(x), x) - y(x))**2 - 3*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)) + 3*y(x))*y(x)) + y(x))/x cannot be solved by the factorable group method