15.8.46 problem 49

Internal problem ID [3049]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 49
Date solved : Sunday, March 30, 2025 at 01:14:03 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

With initial conditions

\begin{align*} y \left (2\right )&=0 \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.198 (sec). Leaf size: 36

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

\begin{align*} y(x)\to -\sqrt {x-2} \sqrt {x} \\ y(x)\to \sqrt {x-2} \sqrt {x} \\ \end{align*}

✓ Sympy. Time used: 0.433 (sec). Leaf size: 22

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

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