47.1.10 problem 10

Internal problem ID [7391]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 04:25:10 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

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

\begin{align*} y &= \sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +2} \\ y &= -\sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +2} \\ \end{align*}

✓ Mathematica. Time used: 0.307 (sec). Leaf size: 70

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

\begin{align*} y(x)\to -\frac {\sqrt {2 x+e^{2 c_1}}}{\sqrt {x}} \\ y(x)\to \frac {\sqrt {2 x+e^{2 c_1}}}{\sqrt {x}} \\ y(x)\to -\sqrt {2} \\ y(x)\to \sqrt {2} \\ \end{align*}

✓ Sympy. Time used: 0.672 (sec). Leaf size: 29

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

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