53.1.5 problem 5

Internal problem ID [8439]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 05:47:51 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (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 &= -\ln \left (x \right )+c_{1} \\ y &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (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: 0.426 (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)
 

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