89.32.14 problem 16

Internal problem ID [24974]
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. Miscellaneous Exercises at page 246
Problem number : 16
Date solved : Thursday, October 02, 2025 at 11:45:14 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Sympy. Time used: 0.154 (sec). Leaf size: 15

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

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