67.7.12 problem 12

Internal problem ID [16457]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 12
Date solved : Thursday, October 02, 2025 at 01:34:01 PM
CAS classification : [_exact, _rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 50

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

\begin{align*} y &= \frac {\sqrt {\left (x^{2}+1\right ) \left (-x +c_1 \right )}}{x^{2}+1} \\ y &= -\frac {\sqrt {\left (x^{2}+1\right ) \left (-x +c_1 \right )}}{x^{2}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.151 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to -\frac {\sqrt {-x+c_1}}{\sqrt {x^2+1}}\\ y(x)&\to \frac {\sqrt {-x+c_1}}{\sqrt {x^2+1}} \end{align*}

✓ Sympy. Time used: 0.402 (sec). Leaf size: 29

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

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