83.47.13 problem Ex 13 page 91

Internal problem ID [19519]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VI. Homogeneous linear equations with variable coefficients
Problem number : Ex 13 page 91
Date solved : Monday, March 31, 2025 at 07:28:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 x^{2} y y^{\prime \prime }+4 y^{2}&=x^{2} {y^{\prime }}^{2}+2 x y y^{\prime } \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 25

ode:=2*x^2*y(x)*diff(diff(y(x),x),x)+4*y(x)^2 = x^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {x^{2} \left (2 c_{2} \ln \left (x \right )+c_{1} \right )^{2}}{4 c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.443 (sec). Leaf size: 20

ode=2*x^2*y[x]*D[y[x],{x,2}]+4*y[x]^2==x^2*D[y[x],x]^2+2*x*y[x]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 x^2 (\log (x)-2 c_1){}^2 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*y(x)*Derivative(y(x), (x, 2)) - x**2*Derivative(y(x), x)**2 - 2*x*y(x)*Derivative(y(x), x) + 4*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((2*x**2*Derivative(y(x), (x, 2)) + 5*y(x))*y(x)) - y(x))/x cannot be solved by the factorable group method