47.2.44 problem Example 5

Internal problem ID [7460]
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 : Example 5
Date solved : Sunday, March 30, 2025 at 12:09:21 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 67

ode:=2*x*diff(y(x),x)+(x^2*y(x)^4+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1}{\sqrt {\sqrt {2 \ln \left (x \right )+c_1}\, x}} \\ y &= \frac {1}{\sqrt {-\sqrt {2 \ln \left (x \right )+c_1}\, x}} \\ y &= -\frac {1}{\sqrt {\sqrt {2 \ln \left (x \right )+c_1}\, x}} \\ y &= -\frac {1}{\sqrt {-\sqrt {2 \ln \left (x \right )+c_1}\, x}} \\ \end{align*}

✓ Mathematica. Time used: 0.614 (sec). Leaf size: 92

ode=2*x*D[y[x],x]+(x^2*y[x]^4+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to -\frac {i}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to \frac {i}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to \frac {1}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 3.860 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + (x**2*y(x)**4 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - i \sqrt [4]{\frac {1}{x^{2} \left (C_{1} + 2 \log {\left (x \right )}\right )}}, \ y{\left (x \right )} = i \sqrt [4]{\frac {1}{x^{2} \left (C_{1} + 2 \log {\left (x \right )}\right )}}, \ y{\left (x \right )} = - \sqrt [4]{\frac {1}{x^{2} \left (C_{1} + 2 \log {\left (x \right )}\right )}}, \ y{\left (x \right )} = \sqrt [4]{\frac {1}{x^{2} \left (C_{1} + 2 \log {\left (x \right )}\right )}}\right ] \]