89.28.18 problem 18

Internal problem ID [24906]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 229
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:49:23 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

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

\begin{align*} y &= c_1 x \\ y &= x +c_1 \\ y &= \frac {x^{2}}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 36

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

\begin{align*} y(x)&\to c_1 x\\ y(x)&\to x+c_1\\ y(x)&\to \frac {x^2}{2}+c_1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 19

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

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