83.34.2 problem 2

Internal problem ID [19344]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (H) at page 118
Problem number : 2
Date solved : Monday, March 31, 2025 at 07:09:06 PM
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 y y^{\prime \prime }+x {y^{\prime }}^{2}&=3 y y^{\prime } \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 39

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

\begin{align*} y &= 0 \\ y &= -\frac {\sqrt {2 c_1 \,x^{4}+8 c_2}}{2} \\ y &= \frac {\sqrt {2 c_1 \,x^{4}+8 c_2}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.274 (sec). Leaf size: 20

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

\[ y(x)\to c_2 \sqrt {x^4+2 c_1} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x)**2 - 3*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)) + 9*y(x))*y(x)) + 3*y(x))/(2*x) cannot be solved by the factorable group method