4.7.11 \(\left (a^2+x^2\right ) y'(x)=a^2-2 y(x)^2+3 x y(x)\)

ODE
\[ \left (a^2+x^2\right ) y'(x)=a^2-2 y(x)^2+3 x y(x) \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.156724 (sec), leaf count = 58

\[\left \{\left \{y(x)\to \frac {a^2 c_1 (-x) \sqrt {a^2+x^2}+a^2+2 x^2}{2 x-a^2 c_1 \sqrt {a^2+x^2}}\right \}\right \}\]

Maple
cpu = 0.367 (sec), leaf count = 280

\[ \left \{ y \relax (x ) =-{\frac {a}{ia+x} \left (-{\frac {\sqrt {2} \left (ix-a \right ) {\it \_C1}}{2}\sqrt {{\frac {ix-a}{a}}}{\it HeunC} \left (0,-{\frac {1}{2}},2,0,{\frac {5}{4}},{\frac {a+ix}{ix-a}} \right ) }+ \left (ix-a \right ) \sqrt {{\frac {a+ix}{a}}}{\it HeunC} \left (0,{\frac {1}{2}},2,0,{\frac {5}{4}},{\frac {a+ix}{ix-a}} \right ) + \left (-{\it \_C1}\,\sqrt {2}\sqrt {{\frac {ix-a}{a}}}{\it HeunCPrime} \left (0,-{\frac {1}{2}},2,0,{\frac {5}{4}},{\frac {a+ix}{ix-a}} \right ) +{\it HeunCPrime} \left (0,{\frac {1}{2}},2,0,{\frac {5}{4}},{\frac {a+ix}{ix-a}} \right ) \sqrt {{\frac {a+ix}{a}}} \right ) \left (a+ix \right ) \right ) \left (\sqrt {2}\sqrt {{\frac {ix-a}{a}}}{\it HeunC} \left (0,-{\frac {1}{2}},2,0,{\frac {5}{4}},{\frac {a+ix}{ix-a}} \right ) {\it \_C1}-{\it HeunC} \left (0,{\frac {1}{2}},2,0,{\frac {5}{4}},{\frac {a+ix}{ix-a}} \right ) \sqrt {{\frac {a+ix}{a}}} \right ) ^{-1}} \right \} \] Mathematica raw input

DSolve[(a^2 + x^2)*y'[x] == a^2 + 3*x*y[x] - 2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (a^2 + 2*x^2 - a^2*x*Sqrt[a^2 + x^2]*C[1])/(2*x - a^2*Sqrt[a^2 + x^2]*
C[1])}}

Maple raw input

dsolve((a^2+x^2)*diff(y(x),x) = a^2+3*x*y(x)-2*y(x)^2, y(x),'implicit')

Maple raw output

y(x) = -a*(-1/2*((I*x-a)/a)^(1/2)*2^(1/2)*(I*x-a)*_C1*HeunC(0,-1/2,2,0,5/4,(a+I*
x)/(I*x-a))+(I*x-a)*((a+I*x)/a)^(1/2)*HeunC(0,1/2,2,0,5/4,(a+I*x)/(I*x-a))+(-_C1
*2^(1/2)*((I*x-a)/a)^(1/2)*HeunCPrime(0,-1/2,2,0,5/4,(a+I*x)/(I*x-a))+HeunCPrime
(0,1/2,2,0,5/4,(a+I*x)/(I*x-a))*((a+I*x)/a)^(1/2))*(a+I*x))/(2^(1/2)*((I*x-a)/a)
^(1/2)*HeunC(0,-1/2,2,0,5/4,(a+I*x)/(I*x-a))*_C1-HeunC(0,1/2,2,0,5/4,(a+I*x)/(I*
x-a))*((a+I*x)/a)^(1/2))/(I*a+x)