77.1.89 problem 116 (page 177)

Internal problem ID [17908]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 116 (page 177)
Date solved : Monday, March 31, 2025 at 04:49:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 23

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

\[ y = -\ln \left (\frac {c_1 x -c_2}{n x}\right ) x n \]

✓ Mathematica. Time used: 8.107 (sec). Leaf size: 25

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

\[ y(x)\to -n x \log \left (-\frac {c_2 x+c_1}{n x}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(n*x**3*Derivative(y(x), (x, 2)) - (-x*Derivative(y(x), x) + y(x))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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