62.9.5 problem Ex 5

Internal problem ID [12757]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 16. Integrating factors by inspection. Page 23
Problem number : Ex 5
Date solved : Wednesday, March 05, 2025 at 08:25:36 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 44

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

\begin{align*} y(x)\to -\sqrt {x} \sqrt {-\log (x)+c_1} \\ y(x)\to \sqrt {x} \sqrt {-\log (x)+c_1} \\ \end{align*}

✓ Sympy. Time used: 0.453 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = - \sqrt {x \left (C_{1} - \log {\left (x \right )}\right )}, \ y{\left (x \right )} = \sqrt {x \left (C_{1} - \log {\left (x \right )}\right )}\right ] \]