15.8.22 problem 23

Internal problem ID [3025]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 23
Date solved : Sunday, March 30, 2025 at 01:10:59 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.180 (sec). Leaf size: 51

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

\[ y = \frac {1}{2}+\frac {\sqrt {3}\, \left (x +1\right ) \tan \left (\operatorname {RootOf}\left (-2 \sqrt {3}\, \ln \left (2\right )+\sqrt {3}\, \ln \left (\left (x +1\right )^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 \sqrt {3}\, c_1 +2 \textit {\_Z} \right )\right )}{6}-\frac {x}{2} \]

✓ Mathematica. Time used: 0.1 (sec). Leaf size: 73

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

\[ \text {Solve}\left [\frac {\arctan \left (\frac {\sqrt {3} (y(x)-1)}{3 y(x)+2 x-1}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\frac {3 \left (x^2+3 y(x)^2+3 (x-1) y(x)-x+1\right )}{(x+1)^2}\right )+\log (x+1)+c_1=0,y(x)\right ] \]

✓ Sympy. Time used: 5.494 (sec). Leaf size: 60

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

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