67.7.22 problem 22

Internal problem ID [16467]
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 : 22
Date solved : Thursday, October 02, 2025 at 01:34:22 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} x y y^{\prime }&=2 x^{2}+2 y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

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

\begin{align*} y &= \sqrt {c_1 \,x^{2}-2}\, x \\ y &= -\sqrt {c_1 \,x^{2}-2}\, x \\ \end{align*}

✓ Mathematica. Time used: 0.343 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 0.235 (sec). Leaf size: 29

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

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