89.3.20 problem 20

Internal problem ID [24318]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 34
Problem number : 20
Date solved : Thursday, October 02, 2025 at 10:17:48 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 53

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

\begin{align*} y &= \frac {3 c_1 x -\sqrt {5 x^{2} c_1^{2}+4}}{2 c_1} \\ y &= \frac {3 c_1 x +\sqrt {5 x^{2} c_1^{2}+4}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.241 (sec). Leaf size: 110

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

✓ Sympy. Time used: 0.833 (sec). Leaf size: 37

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

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