89.28.9 problem 9

Internal problem ID [24897]
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 : 9
Date solved : Thursday, October 02, 2025 at 10:49:11 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 28

ode=x*D[y[x],x]^2+ (1-x^2*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 e^{\frac {x^2}{2}}\\ y(x)&\to -\log (x)+c_1 \end{align*}

✓ Sympy. Time used: 0.245 (sec). Leaf size: 17

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

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