4.13.6 \(\left (-x^2+y(x)^2+1\right ) y'(x)=x^2-y(x)^2+1\)

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

\[ \left \{ {\frac { \left (y \relax (x ) \right ) ^{2}}{2}}+xy \relax (x ) +\ln \left (y \relax (x ) -x \right ) +{\frac {{x}^{2}}{2}}-{\it \_C1}=0 \right \} \] 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