54.1.254 problem 259

Internal problem ID [11568]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 259
Date solved : Tuesday, September 30, 2025 at 09:20:43 PM
CAS classification : [_Bernoulli]

\begin{align*} 2 x^{2} y y^{\prime }-y^{2}-x^{2} {\mathrm e}^{x -\frac {1}{x}}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

\begin{align*} y &= \sqrt {{\mathrm e}^{-\frac {1}{x}} \left (c_1 +{\mathrm e}^{x}\right )} \\ y &= -\sqrt {{\mathrm e}^{-\frac {1}{x}} \left (c_1 +{\mathrm e}^{x}\right )} \\ \end{align*}

✓ Mathematica. Time used: 1.089 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to -e^{\left .-\frac {1}{2}\right /x} \sqrt {e^x+c_1}\\ y(x)&\to e^{\left .-\frac {1}{2}\right /x} \sqrt {e^x+c_1} \end{align*}

✓ Sympy. Time used: 0.775 (sec). Leaf size: 39

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

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