41.2.53 problem 49

Internal problem ID [8755]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 49
Date solved : Tuesday, September 30, 2025 at 05:50:16 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 16

ode:=2*y(x)+(x^2*y(x)+1)*x*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{\operatorname {LambertW}\left (\frac {c_1}{x^{2}}\right ) x^{2}} \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 73

ode=2*y[x]+(x^2*y[x]+1)*x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {(-1)^{2/3} \left (\frac {6}{y(x) x^2+1}-4\right )}{2 \sqrt [3]{2}}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]+\frac {2}{9} (-2)^{2/3} \log (x)=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.456 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2*y(x) + 1)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{C_{1} + W\left (\frac {e^{- C_{1}}}{x^{2}}\right )} \]