82.8.2 problem Ex. 2

Internal problem ID [18679]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 25
Problem number : Ex. 2
Date solved : Monday, March 31, 2025 at 05:56:21 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} x^{2}+y^{2}-x^{2} y y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 67

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

\begin{align*} y &= \sqrt {4 \,\operatorname {Ei}_{1}\left (-\frac {2}{x}\right ) {\mathrm e}^{-\frac {2}{x}}+{\mathrm e}^{-\frac {2}{x}} c_1 +2 x} \\ y &= -\sqrt {4 \,\operatorname {Ei}_{1}\left (-\frac {2}{x}\right ) {\mathrm e}^{-\frac {2}{x}}+{\mathrm e}^{-\frac {2}{x}} c_1 +2 x} \\ \end{align*}

✓ Mathematica. Time used: 2.029 (sec). Leaf size: 76

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

\begin{align*} y(x)\to -e^{-1/x} \sqrt {-4 \operatorname {ExpIntegralEi}\left (\frac {2}{x}\right )+2 e^{2/x} x+c_1} \\ y(x)\to e^{-1/x} \sqrt {-4 \operatorname {ExpIntegralEi}\left (\frac {2}{x}\right )+2 e^{2/x} x+c_1} \\ \end{align*}

✓ Sympy. Time used: 1.368 (sec). Leaf size: 56

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

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