69.4.18 problem 63

Internal problem ID [18007]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 63
Date solved : Thursday, October 02, 2025 at 02:33:10 PM
CAS classification : [_separable]

\begin{align*} a^{2}+y^{2}+2 x \sqrt {a x -x^{2}}\, y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.159 (sec). Leaf size: 22

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

\[ y = \tan \left (\frac {a -x}{\sqrt {x \left (a -x \right )}}\right ) a \]

✗ Mathematica

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

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✗ Sympy

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

ValueError : Couldnt solve for initial conditions