4.1.40 \(y'(x)=x^2-y(x)^2\)

ODE
\[ y'(x)=x^2-y(x)^2 \] ODE Classification

[_Riccati]

Book solution method
Series solution to \(y'(x)=f(x,y(x))\), case \(f(x,y)\) analytic

Mathematica
cpu = 0.00687899 (sec), leaf count = 117

\[\left \{\left \{y(x)\to -\frac {-i x^2 \left (c_1 \left (J_{-\frac {5}{4}}\left (\frac {i x^2}{2}\right )-J_{\frac {3}{4}}\left (\frac {i x^2}{2}\right )\right )+2 J_{-\frac {3}{4}}\left (\frac {i x^2}{2}\right )\right )-c_1 J_{-\frac {1}{4}}\left (\frac {i x^2}{2}\right )}{2 x \left (c_1 J_{-\frac {1}{4}}\left (\frac {i x^2}{2}\right )+J_{\frac {1}{4}}\left (\frac {i x^2}{2}\right )\right )}\right \}\right \}\]

Maple
cpu = 0.609 (sec), leaf count = 44

\[ \left \{ y \relax (x ) ={x \left ({{\sl I}_{-{\frac {3}{4}}}\left ({\frac {{x}^{2}}{2}}\right )}{\it \_C1}-{{\sl K}_{{\frac {3}{4}}}\left ({\frac {{x}^{2}}{2}}\right )} \right ) \left ({\it \_C1}\,{{\sl I}_{{\frac {1}{4}}}\left ({\frac {{x}^{2}}{2}}\right )}+{{\sl K}_{{\frac {1}{4}}}\left ({\frac {{x}^{2}}{2}}\right )} \right ) ^{-1}} \right \} \] Mathematica raw input

DSolve[y'[x] == x^2 - y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -(-(BesselJ[-1/4, (I/2)*x^2]*C[1]) - I*x^2*(2*BesselJ[-3/4, (I/2)*x^2]
 + (BesselJ[-5/4, (I/2)*x^2] - BesselJ[3/4, (I/2)*x^2])*C[1]))/(2*x*(BesselJ[1/4
, (I/2)*x^2] + BesselJ[-1/4, (I/2)*x^2]*C[1]))}}

Maple raw input

dsolve(diff(y(x),x) = x^2-y(x)^2, y(x),'implicit')

Maple raw output

y(x) = x*(BesselI(-3/4,1/2*x^2)*_C1-BesselK(3/4,1/2*x^2))/(_C1*BesselI(1/4,1/2*x
^2)+BesselK(1/4,1/2*x^2))