89.8.16 problem 18

Internal problem ID [24476]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 66
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:40:55 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

\begin{align*} x -1-\left (3 x -2 y-5\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 1.319 (sec). Leaf size: 30

ode:=x-1-(3*x-2*y(x)-5)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-1-\sqrt {1+\left (8 x -8\right ) c_1}+4 \left (-3+x \right ) c_1}{8 c_1} \]

✓ Mathematica. Time used: 8.4 (sec). Leaf size: 303

ode=( x-1)-( 3*x-2*y[x]-5 )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (3 x-\frac {(2-2 i) (x-1)}{\frac {i \sqrt {2}}{\sqrt {(x-1) \cosh \left (\frac {2 c_1}{9}\right )+(x-1) \sinh \left (\frac {2 c_1}{9}\right )-i}}+(1-i)}-5\right )\\ y(x)&\to \frac {1}{2} \left (3 x-\frac {(2-2 i) (x-1)}{(1-i)-\frac {i \sqrt {2}}{\sqrt {(x-1) \cosh \left (\frac {2 c_1}{9}\right )+(x-1) \sinh \left (\frac {2 c_1}{9}\right )-i}}}-5\right )\\ y(x)&\to \frac {1}{2} \left (3 x+\frac {(2-2 i) (x-1) \sqrt {(x-1) \cosh \left (\frac {2 c_1}{9}\right )+(x-1) \sinh \left (\frac {2 c_1}{9}\right )+i}}{\sqrt {2}-(1-i) \sqrt {(x-1) \cosh \left (\frac {2 c_1}{9}\right )+(x-1) \sinh \left (\frac {2 c_1}{9}\right )+i}}-5\right )\\ y(x)&\to \frac {1}{2} \left (3 x-\frac {(2-2 i) (x-1)}{\frac {\sqrt {2}}{\sqrt {(x-1) \cosh \left (\frac {2 c_1}{9}\right )+(x-1) \sinh \left (\frac {2 c_1}{9}\right )+i}}+(1-i)}-5\right )\\ y(x)&\to \frac {x-3}{2}\\ y(x)&\to x-2 \end{align*}

✓ Sympy. Time used: 1.587 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x - (3*x - 2*y(x) - 5)*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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