77.1.97 problem 124 (page 179)

Internal problem ID [17908]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 124 (page 179)
Date solved : Friday, March 14, 2025 at 12:41:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 x y y^{\prime }&=4 y^{2} \end{align*}

✓ Maple. Time used: 0.109 (sec). Leaf size: 87

ode:=x^2*y(x)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)^2-5*diff(y(x),x)*x*y(x) = 4*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= -\frac {x^{3-\sqrt {17}} 17^{{3}/{4}} \sqrt {-\left (x^{2 \sqrt {17}} c_{1} -c_{2} \right ) x^{-3+\sqrt {17}}}}{17} \\ y &= \frac {x^{3-\sqrt {17}} 17^{{3}/{4}} \sqrt {-\left (x^{2 \sqrt {17}} c_{1} -c_{2} \right ) x^{-3+\sqrt {17}}}}{17} \\ \end{align*}

✓ Mathematica. Time used: 2.899 (sec). Leaf size: 39

ode=x^2*y[x]*D[y[x],{x,2}]+x^2*D[y[x],x]^2-5*x*y[x]*D[y[x],x]==4*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 x^{\frac {3}{2}-\frac {\sqrt {17}}{2}} \sqrt {x^{2 \sqrt {17}}+c_1} \]

✗ 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 - 5*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((-4*x**2*Derivative(y(x), (x, 2)) + 41*y(x))*y(x)) + 5*y(x))/(2*x) cannot be solved by the factorable group method