62.9.3 problem Ex 3

Internal problem ID [12763]
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 3
Date solved : Monday, March 31, 2025 at 07:03:18 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 24

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

\[ y = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_1 \right )\right ) x \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 36

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]-1}{K[1]^2+1}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.376 (sec). Leaf size: 26

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

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