53.1.10 problem 10

Internal problem ID [8444]
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 : 10
Date solved : Wednesday, March 05, 2025 at 05:48:00 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

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

\begin{align*} y &= \sqrt {-x^{2}+c_{1}} \\ y &= -\sqrt {-x^{2}+c_{1}} \\ y &= {\mathrm e}^{x} c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 54

ode=y[x]*(D[y[x],x])^2+(x-y[x]^2)*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^x \\ y(x)\to -\sqrt {-x^2+2 c_1} \\ y(x)\to \sqrt {-x^2+2 c_1} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.599 (sec). Leaf size: 29

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

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