56.1.86 problem 85

Internal problem ID [8798]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 85
Date solved : Sunday, March 30, 2025 at 01:37:39 PM
CAS classification : [_Bernoulli]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 9

ode:=diff(y(x),x) = 2*y(x)*(x*y(x)^(1/2)-1); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.682 (sec). Leaf size: 20

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

\begin{align*} y(x)\to 1 \\ y(x)\to \sec ^2\left (\frac {x^2}{2}\right ) \\ \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants