ODE
\[ \left (y(x)^2+1\right ) y''(x)=(a+3 y(x)) y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.557184 (sec), leaf count = 47
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 4.489 (sec), leaf count = 375
\[[y \left (x \right ) = \tan \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{4} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{2} x^{2}+{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C2}^{2} a^{4}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \,a^{3} x +4 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{2} x -2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C2} \,a^{3}+{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C2}^{2} a^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} a x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1} \textit {\_C2} x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C2} a +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C2}^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right ), y \left (x \right ) = \tan \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{4} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{2} x^{2}+{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C2}^{2} a^{4}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \,a^{3} x +4 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{2} x +2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C2} \,a^{3}+{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C2}^{2} a^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} a x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1} \textit {\_C2} x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C2} a +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C2}^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )]\] Mathematica raw input
DSolve[(1 + y[x]^2)*y''[x] == (a + 3*y[x])*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-a + #1)/((1 + a^2)*E^(a*ArcTan[#1])*C[1]*Sqrt[1 + #1^2]) & ][x + C[2]]}}
Maple raw input
dsolve((1+y(x)^2)*diff(diff(y(x),x),x) = (a+3*y(x))*diff(y(x),x)^2, y(x))
Maple raw output
[y(x) = tan(RootOf(exp(a*_Z)^2*_C1^2*a^4*x^2+2*exp(a*_Z)^2*_C1*_C2*a^4*x+2*exp(a*_Z)^2*_C1^2*a^2*x^2+exp(a*_Z)^2*_C2^2*a^4-2*cos(_Z)*exp(a*_Z)*_C1*a^3*x+4*exp(a*_Z)^2*_C1*_C2*a^2*x-2*cos(_Z)*exp(a*_Z)*_C2*a^3+exp(a*_Z)^2*_C1^2*x^2+2*exp(a*_Z)^2*_C2^2*a^2-2*cos(_Z)*exp(a*_Z)*_C1*a*x+2*exp(a*_Z)^2*_C1*_C2*x+cos(_Z)^2*a^2-2*cos(_Z)*exp(a*_Z)*_C2*a+exp(a*_Z)^2*_C2^2+cos(_Z)^2-1)), y(x) = tan(RootOf(exp(a*_Z)^2*_C1^2*a^4*x^2+2*exp(a*_Z)^2*_C1*_C2*a^4*x+2*exp(a*_Z)^2*_C1^2*a^2*x^2+exp(a*_Z)^2*_C2^2*a^4+2*cos(_Z)*exp(a*_Z)*_C1*a^3*x+4*exp(a*_Z)^2*_C1*_C2*a^2*x+2*cos(_Z)*exp(a*_Z)*_C2*a^3+exp(a*_Z)^2*_C1^2*x^2+2*exp(a*_Z)^2*_C2^2*a^2+2*cos(_Z)*exp(a*_Z)*_C1*a*x+2*exp(a*_Z)^2*_C1*_C2*x+cos(_Z)^2*a^2+2*cos(_Z)*exp(a*_Z)*_C2*a+exp(a*_Z)^2*_C2^2+cos(_Z)^2-1))]