54.1.437 problem 449

Internal problem ID [11751]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 449
Date solved : Tuesday, September 30, 2025 at 10:16:34 PM
CAS classification : [_separable]

\begin{align*} \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=(-a^2+x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c_1}{a -x} \\ y &= \frac {c_1}{a +x} \\ \end{align*}

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to \frac {c_1}{a-x}\\ y(x)&\to \frac {c_1}{a+x}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.219 (sec). Leaf size: 14

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

\[ \left [ y{\left (x \right )} = \frac {C_{1}}{a + x}, \ y{\left (x \right )} = \frac {C_{1}}{- a + x}\right ] \]