64.7.9 problem 9

Internal problem ID [13304]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Section 2.4. Special integrating factors and transformations. Exercises page 67
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 09:41:55 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.109 (sec). Leaf size: 31

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

\[ y = -1-\tan \left (\operatorname {RootOf}\left (-4 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -1\right )+2 c_{1} \right )\right ) \left (x -1\right ) \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 66

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

\[ \text {Solve}\left [32 \arctan \left (\frac {2 y(x)-x+3}{y(x)+2 x-1}\right )+8 \log \left (\frac {x^2+y(x)^2+2 y(x)-2 x+2}{5 (x-1)^2}\right )+16 \log (x-1)+5 c_1=0,y(x)\right ] \]

✓ Sympy. Time used: 3.297 (sec). Leaf size: 36

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

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