56.1.5 problem Ex 5

Internal problem ID [14083]
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 8. Exact differential equations. Page 11
Problem number : Ex 5
Date solved : Thursday, October 02, 2025 at 09:11:00 AM
CAS classification : [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 6 x -2 y+1+\left (2 y-2 x -3\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.170 (sec). Leaf size: 33

ode:=6*x-2*y(x)+1+(2*y(x)-2*x-3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\sqrt {1-8 \left (x -\frac {1}{2}\right )^{2} c_1^{2}}+\left (2 x +3\right ) c_1}{2 c_1} \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 67

ode=(6*x-2*y[x]+1)+(2*y[x]-2*x-3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} i \sqrt {8 x^2-8 x-9-4 c_1}+x+\frac {3}{2}\\ y(x)&\to \frac {1}{2} i \sqrt {8 x^2-8 x-9-4 c_1}+x+\frac {3}{2} \end{align*}

✓ Sympy. Time used: 1.591 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x + (-2*x + 2*y(x) - 3)*Derivative(y(x), x) - 2*y(x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x - \frac {\sqrt {C_{1} - 8 x^{2} + 8 x}}{2} + \frac {3}{2}, \ y{\left (x \right )} = x + \frac {\sqrt {C_{1} - 8 x^{2} + 8 x}}{2} + \frac {3}{2}\right ] \]