89.10.1 problem 1

Internal problem ID [24494]
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 77
Problem number : 1
Date solved : Thursday, October 02, 2025 at 10:42:42 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {3}{2}-\frac {{\mathrm e}^{-x} \sqrt {{\mathrm e}^{2 x} \left (4 x +5\right )-4 \,{\mathrm e}^{x} c_1}}{2} \\ y &= \frac {3}{2}+\frac {{\mathrm e}^{-x} \sqrt {{\mathrm e}^{2 x} \left (4 x +5\right )-4 \,{\mathrm e}^{x} c_1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 12.765 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to \frac {3}{2}-\frac {\sqrt {e^x (4 x+5)+4 c_1}}{2 \sqrt {e^x}}\\ y(x)&\to \frac {1}{2} \left (3+\frac {\sqrt {e^x (4 x+5)+4 c_1}}{\sqrt {e^x}}\right ) \end{align*}

✓ Sympy. Time used: 0.948 (sec). Leaf size: 65

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

\[ \left [ y{\left (x \right )} = \frac {\left (- \sqrt {\left (C_{1} + 4 x e^{x} + 5 e^{x}\right ) e^{x}} + 3 e^{x}\right ) e^{- x}}{2}, \ y{\left (x \right )} = \frac {\left (\sqrt {\left (C_{1} + 4 x e^{x} + 5 e^{x}\right ) e^{x}} + 3 e^{x}\right ) e^{- x}}{2}\right ] \]