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 = 0.22425 (sec), leaf count = 117
\[\left \{\left \{y(x)\to -\frac {-i x^2 \left (2 J_{-\frac {3}{4}}\left (\frac {i x^2}{2}\right )+c_1 \left (J_{-\frac {5}{4}}\left (\frac {i x^2}{2}\right )-J_{\frac {3}{4}}\left (\frac {i x^2}{2}\right )\right )\right )-c_1 J_{-\frac {1}{4}}\left (\frac {i x^2}{2}\right )}{2 x \left (J_{\frac {1}{4}}\left (\frac {i x^2}{2}\right )+c_1 J_{-\frac {1}{4}}\left (\frac {i x^2}{2}\right )\right )}\right \}\right \}\]
Maple ✓
cpu = 0.121 (sec), leaf count = 44
\[\left [y \left (x \right ) = \frac {x \left (\BesselI \left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \textit {\_C1} -\BesselK \left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{\textit {\_C1} \BesselI \left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\BesselK \left (\frac {1}{4}, \frac {x^{2}}{2}\right )}\right ]\] Mathematica raw input
DSolve[y'[x] == x^2 - y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -1/2*(-(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]))/(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))
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))]