54.2.13 problem 15

Internal problem ID [8545]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:17:21 PM
CAS classification : [_quadrature]

\begin{align*} x {y^{\prime }}^{2}-\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: 0.003 (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: 0.250 (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)
 

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