49.21.14 problem 5(b)

Internal problem ID [7744]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 190
Problem number : 5(b)
Date solved : Sunday, March 30, 2025 at 12:22:22 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {2 x +3 y+1}{x -2 y-1} \end{align*}

✓ Maple. Time used: 0.176 (sec). Leaf size: 61

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

\[ y = -\frac {5}{14}+\frac {\sqrt {3}\, \left (7 x -1\right ) \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\left (7 x -1\right )^{2} \sec \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (2\right )+\sqrt {3}\, \ln \left (3\right )+2 \sqrt {3}\, c_1 -4 \textit {\_Z} \right )\right )}{14}-\frac {x}{2} \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 85

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

\[ \text {Solve}\left [32 \sqrt {3} \arctan \left (\frac {4 y(x)+5 x+1}{\sqrt {3} (-2 y(x)+x-1)}\right )=3 \left (8 \log \left (\frac {4 \left (7 x^2+7 y(x)^2+(7 x+5) y(x)+x+1\right )}{(1-7 x)^2}\right )+16 \log (7 x-1)+7 c_1\right ),y(x)\right ] \]

✓ Sympy. Time used: 6.362 (sec). Leaf size: 73

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

\[ \log {\left (x - \frac {1}{7} \right )} = C_{1} - \log {\left (\sqrt {1 + \frac {y{\left (x \right )} + \frac {3}{7}}{x - \frac {1}{7}} + \frac {\left (y{\left (x \right )} + \frac {3}{7}\right )^{2}}{\left (x - \frac {1}{7}\right )^{2}}} \right )} + \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (1 + \frac {2 \left (y{\left (x \right )} + \frac {3}{7}\right )}{x - \frac {1}{7}}\right )}{3} \right )}}{3} \]