29.11.17 problem 308

Internal problem ID [4908]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 11
Problem number : 308
Date solved : Tuesday, March 04, 2025 at 07:26:55 PM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (a^{2}+x^{2}\right ) y^{\prime }&=a^{2}+3 x y-2 y^{2} \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 217

ode:=(a^2+x^2)*diff(y(x),x) = a^2+3*x*y(x)-2*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = -\frac {2 a \left (c_{1} \left (i a -x \right ) a^{2} \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \operatorname {HeunCPrime}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\left (i a -x \right ) a^{2} \sqrt {\frac {i x +a}{a}}\, \operatorname {HeunCPrime}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\frac {c_{1} \sqrt {2}\, \left (i a x -\frac {1}{2} a^{2}+\frac {1}{2} x^{2}\right ) x \sqrt {\frac {i x -a}{a}}}{2}-\frac {\sqrt {\frac {i x +a}{a}}\, \left (i a^{3}-3 i x^{2} a +3 x \,a^{2}-x^{3}\right )}{4}\right )}{\left (i \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, c_{1} x +\frac {\sqrt {\frac {i x +a}{a}}\, \left (i x -a \right )}{2}\right ) \left (i a +x \right )^{2}} \]

✓ Mathematica. Time used: 1.231 (sec). Leaf size: 63

ode=(a^2+x^2)D[y[x],x]==a^2+3 x y[x]-2 y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {a^2 c_1 (-x) \sqrt {a^2+x^2}+a^2+2 x^2}{2 x-a^2 c_1 \sqrt {a^2+x^2}} \\ y(x)\to x \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2 - 3*x*y(x) + (a**2 + x**2)*Derivative(y(x), x) + 2*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out