8.5.10 problem 10

Internal problem ID [738]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 11:37:34 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} x y y^{\prime }&=x^{2}+3 y^{2} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 33

ode:=x*y(x)*diff(y(x),x) = x^2+3*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {4 c_1 \,x^{4}-2}\, x}{2} \\ y &= \frac {\sqrt {4 c_1 \,x^{4}-2}\, x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.604 (sec). Leaf size: 42

ode=x*y[x]*D[y[x],x] == x^2+3*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -x \sqrt {-\frac {1}{2}+c_1 x^4} \\ y(x)\to x \sqrt {-\frac {1}{2}+c_1 x^4} \\ \end{align*}

✓ Sympy. Time used: 0.411 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + x*y(x)*Derivative(y(x), x) - 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {x \sqrt {C_{1} x^{4} - 2}}{2}, \ y{\left (x \right )} = \frac {x \sqrt {C_{1} x^{4} - 2}}{2}\right ] \]