6.5.16 problem 12

Internal problem ID [1640]
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 : 12
Date solved : Tuesday, September 30, 2025 at 04:41:03 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

With initial conditions

\begin{align*} y \left (1\right )&=\frac {\sqrt {2}}{2} \\ \end{align*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 26

ode:=x^2*diff(y(x),x)+2*x*y(x) = y(x)^3; 
ic:=[y(1) = 1/2*2^(1/2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {10}\, \sqrt {4 x^{6}+x}}{8 x^{5}+2} \]

✓ Mathematica. Time used: 0.418 (sec). Leaf size: 29

ode=x^2*D[y[x],x]+2*x*y[x]==y[x]^3; 
ic=y[1]==1/Sqrt[2]; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {\frac {5}{2}} \sqrt {x}}{\sqrt {4 x^5+1}} \end{align*}

✓ Sympy. Time used: 0.447 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + 2*x*y(x) - y(x)**3,0) 
ics = {y(1): sqrt(2)/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {5} \sqrt {\frac {x}{8 x^{5} + 2}} \]