29.33.24 problem 987

Internal problem ID [5563]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 987
Date solved : Sunday, March 30, 2025 at 08:58:49 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.515 (sec). Leaf size: 77

ode:=((-a^2+1)*x^2+y(x)^2)*diff(y(x),x)^2+2*a^2*x*y(x)*diff(y(x),x)+x^2+(-a^2+1)*y(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^{2}-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^{2}-1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_1 \right )\right ) x \\ \end{align*}

✓ Mathematica. Time used: 0.405 (sec). Leaf size: 115

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

\begin{align*} \text {Solve}\left [\sqrt {a-1} \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} \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**2*x*y(x)*Derivative(y(x), x) + x**2 + (1 - a**2)*y(x)**2 + (x**2*(1 - a**2) + y(x)**2)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded