ODE
\[ \left (-x^2+y(x)^2+1\right ) y'(x)=x^2-y(x)^2+1 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 1.64239 (sec), leaf count = 25
\[\text {Solve}\left [e^{\frac {1}{2} (y(x)+x)^2} (x-y(x))=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.037 (sec), leaf count = 29
\[ \left \{ {\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2}}+xy \left ( x \right ) +\ln \left ( y \left ( x \right ) -x \right ) +{\frac {{x}^{2}}{2}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[(1 - x^2 + y[x]^2)*y'[x] == 1 + x^2 - y[x]^2,y[x],x]
Mathematica raw output
Solve[E^((x + y[x])^2/2)*(x - y[x]) == C[1], y[x]]
Maple raw input
dsolve((1-x^2+y(x)^2)*diff(y(x),x) = 1+x^2-y(x)^2, y(x),'implicit')
Maple raw output
1/2*y(x)^2+x*y(x)+ln(y(x)-x)+1/2*x^2-_C1 = 0