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 = 0.646663 (sec), leaf count = 187
\[\left \{\left \{y(x)\to c_3-\frac {\left (1-i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right ){}^{-\frac {1}{2}-\frac {i a}{2}} \left (1+i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right ){}^{\frac {1}{2} i (a+i)} \left (1+a \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right )}{\left (a^2+1\right ) c_1}\right \}\right \}\]
Maple ✓
cpu = 6.796 (sec), leaf count = 789
\[\left [y \left (x \right ) = \int \frac {\sin \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{4}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{4} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{2}+4 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{2} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{2} x^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{3}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \,a^{3} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} x^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} a -2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} a x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )}{\cos \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{4}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{4} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{2}+4 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{2} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{2} x^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{3}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \,a^{3} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} x^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} a -2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} a x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )}d x +\textit {\_C3}, y \left (x \right ) = \int \frac {\sin \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{4}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{4} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{2}+4 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{2} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{2} x^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{3}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \,a^{3} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} x^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} a +2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} a x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )}{\cos \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{4}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{4} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2} a^{2}+4 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} \,a^{2} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} a^{2} x^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} \,a^{3}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \,a^{3} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2}^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} \textit {\_C2} x +{\mathrm e}^{2 a \textit {\_Z}} \textit {\_C1}^{2} x^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} \textit {\_C2} a +2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} \textit {\_C1} a x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )}d x +\textit {\_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
{{y[x] -> C[3] - ((1 - I*InverseFunction[(-a + #1)/((1 + a^2)*E^(a*ArcTan[#1])*C[1]*Sqrt[1 + #1^2]) & ][x + C[2]])^(-1/2 - (I/2)*a)*(1 + I*InverseFunction[(-a + #1)/((1 + a^2)*E^(a*ArcTan[#1])*C[1]*Sqrt[1 + #1^2]) & ][x + C[2]])^((I/2)*(I + a))*(1 + a*InverseFunction[(-a + #1)/((1 + a^2)*E^(a*ArcTan[#1])*C[1]*Sqrt[1 + #1^2]) & ][x + C[2]]))/((1 + a^2)*C[1])}}
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))
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*cos(_Z)*exp(a*_Z)*_C1*_C2*a^3-2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a-2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a^3-2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a-2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a^3+2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a+2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a^3+2*cos(_Z)*exp(a*_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*cos(_Z)*exp(a*_Z)*_C1*_C2*a+2*cos(_Z)*exp(a*_Z)*_C1*a*x+cos(_Z)^2*a^2+cos(_Z)^2-1)),x)+_C3]