54.1.482 problem 495

Internal problem ID [11796]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 495
Date solved : Tuesday, September 30, 2025 at 10:37:17 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left (y^{2}+\left (1-a \right ) x^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 75

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a -1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_1 \right )\right ) x \\ y &= \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a -1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_1 \right )\right ) x \\ \end{align*}

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 101

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

\begin{align*} \text {Solve}\left [\sqrt {a-1} \arctan \left (\frac {y(x)}{x}\right )-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=\log (x)+c_1,y(x)\right ]\\ \text {Solve}\left [\sqrt {a-1} \arctan \left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ]\\ y(x)&\to -i x\\ y(x)&\to i x \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded