15.7.22 problem 22

Internal problem ID [3003]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 22
Date solved : Sunday, March 30, 2025 at 01:05:19 AM
CAS classification : [_rational, _Bernoulli]

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

With initial conditions

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

✓ Maple. Time used: 0.381 (sec). Leaf size: 43

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

\[ y = \left (\frac {4}{9}-\frac {4 i}{9}\right ) \left (x +1\right )^{{5}/{4}} \left (x -1\right )^{{5}/{4}} \sqrt {2}+\frac {x^{4}}{9}-\frac {16 i \sqrt {x -1}\, \sqrt {x +1}}{9}-\frac {2 x^{2}}{9}+\frac {1}{9} \]

✓ Mathematica. Time used: 0.27 (sec). Leaf size: 130

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

\begin{align*} y(x)\to \frac {1}{9} \left (x^4+\left (4 (-1)^{3/4} \sqrt [4]{x^2-1}-2\right ) x^2-4 i \sqrt {x^2-1}-4 (-1)^{3/4} \sqrt [4]{x^2-1}+1\right ) \\ y(x)\to \frac {1}{9} \left (x^4-2 \left (4 (-1)^{3/4} \sqrt [4]{x^2-1}+1\right ) x^2-16 i \sqrt {x^2-1}+8 (-1)^{3/4} \sqrt [4]{x^2-1}+1\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(1 - x**2)*sqrt(y(x)) + x*y(x) + (1 - x**2)*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