ODE
\[ \left (y'(x)^2+1\right ) y'''(x)=y''(x)^2 \left (a+3 y'(x)\right ) \] ODE Classification
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]
Book solution method
TO DO
Mathematica ✗
cpu = 599.996 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 1.585 (sec), leaf count = 789
\[ \left \{ y \left ( x \right ) =\int \!{\frac {\sin \left ( {\it RootOf} \left ( {{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{2}{x}^{2}-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,{a}^{3}-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{x}^{2}-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,a-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}-1 \right ) \right ) }{\cos \left ( {\it RootOf} \left ( {{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{2}{x}^{2}-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,{a}^{3}-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{x}^{2}-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,a-2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}-1 \right ) \right ) }}\,{\rm d}x+{\it \_C3},y \left ( x \right ) =\int \!{\frac {\sin \left ( {\it RootOf} \left ( {{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{2}{x}^{2}+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,{a}^{3}+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{x}^{2}+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,a+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}-1 \right ) \right ) }{\cos \left ( {\it RootOf} \left ( {{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{a}^{2}{x}^{2}+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,{a}^{3}+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,a{\it \_Z}}}{{\it \_C1}}^{2}{x}^{2}+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,{\it \_C2}\,a+2\,{{\rm e}^{a{\it \_Z}}}\cos \left ( {\it \_Z} \right ) {\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}-1 \right ) \right ) }}\,{\rm d}x+{\it \_C3} \right \} \] Mathematica raw input
DSolve[(1 + y'[x]^2)*y'''[x] == (a + 3*y'[x])*y''[x]^2,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve((1+diff(y(x),x)^2)*diff(diff(diff(y(x),x),x),x) = (a+3*diff(y(x),x))*diff(diff(y(x),x),x)^2, y(x),'implicit')
Maple raw output
y(x) = Int(sin(RootOf(exp(2*a*_Z)*_C1^2*_C2^2*a^4+2*exp(2*a*_Z)*_C1^2*_C2*a^4*x+exp(2*a*_Z)*_C1^2*a^4*x^2+2*exp(2*a*_Z)*_C1^2*_C2^2*a^2+4*exp(2*a*_Z)*_C1^2*_C2*a^2*x+2*exp(2*a*_Z)*_C1^2*a^2*x^2-2*exp(a*_Z)*cos(_Z)*_C1*_C2*a^3-2*exp(a*_Z)*cos(_Z)*_C1*a^3*x+exp(2*a*_Z)*_C1^2*_C2^2+2*exp(2*a*_Z)*_C1^2*_C2*x+exp(2*a*_Z)*_C1^2*x^2-2*exp(a*_Z)*cos(_Z)*_C1*_C2*a-2*exp(a*_Z)*cos(_Z)*_C1*a*x+cos(_Z)^2*a^2+cos(_Z)^2-1))/cos(RootOf(exp(2*a*_Z)*_C1^2*_C2^2*a^4+2*exp(2*a*_Z)*_C1^2*_C2*a^4*x+exp(2*a*_Z)*_C1^2*a^4*x^2+2*exp(2*a*_Z)*_C1^2*_C2^2*a^2+4*exp(2*a*_Z)*_C1^2*_C2*a^2*x+2*exp(2*a*_Z)*_C1^2*a^2*x^2-2*exp(a*_Z)*cos(_Z)*_C1*_C2*a^3-2*exp(a*_Z)*cos(_Z)*_C1*a^3*x+exp(2*a*_Z)*_C1^2*_C2^2+2*exp(2*a*_Z)*_C1^2*_C2*x+exp(2*a*_Z)*_C1^2*x^2-2*exp(a*_Z)*cos(_Z)*_C1*_C2*a-2*exp(a*_Z)*cos(_Z)*_C1*a*x+cos(_Z)^2*a^2+cos(_Z)^2-1)),x)+_C3, y(x) = Int(sin(RootOf(exp(2*a*_Z)*_C1^2*_C2^2*a^4+2*exp(2*a*_Z)*_C1^2*_C2*a^4*x+exp(2*a*_Z)*_C1^2*a^4*x^2+2*exp(2*a*_Z)*_C1^2*_C2^2*a^2+4*exp(2*a*_Z)*_C1^2*_C2*a^2*x+2*exp(2*a*_Z)*_C1^2*a^2*x^2+2*exp(a*_Z)*cos(_Z)*_C1*_C2*a^3+2*exp(a*_Z)*cos(_Z)*_C1*a^3*x+exp(2*a*_Z)*_C1^2*_C2^2+2*exp(2*a*_Z)*_C1^2*_C2*x+exp(2*a*_Z)*_C1^2*x^2+2*exp(a*_Z)*cos(_Z)*_C1*_C2*a+2*exp(a*_Z)*cos(_Z)*_C1*a*x+cos(_Z)^2*a^2+cos(_Z)^2-1))/cos(RootOf(exp(2*a*_Z)*_C1^2*_C2^2*a^4+2*exp(2*a*_Z)*_C1^2*_C2*a^4*x+exp(2*a*_Z)*_C1^2*a^4*x^2+2*exp(2*a*_Z)*_C1^2*_C2^2*a^2+4*exp(2*a*_Z)*_C1^2*_C2*a^2*x+2*exp(2*a*_Z)*_C1^2*a^2*x^2+2*exp(a*_Z)*cos(_Z)*_C1*_C2*a^3+2*exp(a*_Z)*cos(_Z)*_C1*a^3*x+exp(2*a*_Z)*_C1^2*_C2^2+2*exp(2*a*_Z)*_C1^2*_C2*x+exp(2*a*_Z)*_C1^2*x^2+2*exp(a*_Z)*cos(_Z)*_C1*_C2*a+2*exp(a*_Z)*cos(_Z)*_C1*a*x+cos(_Z)^2*a^2+cos(_Z)^2-1)),x)+_C3