6.5.23 problem 20

Internal problem ID [1647]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 04:41:19 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

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

\begin{align*} y &= \sqrt {c_1 \,x^{2}-1}\, x \\ y &= -\sqrt {c_1 \,x^{2}-1}\, x \\ \end{align*}

✓ Mathematica. Time used: 0.322 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 0.231 (sec). Leaf size: 29

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

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