ODE
\[ 3 y'(x)^5-y(x) y'(x)+1=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 0.250505 (sec), leaf count = 176
\[\left \{\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[1] \text {$\#$1}+1\& ,1\right ]}dK[1],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[2] \text {$\#$1}+1\& ,2\right ]}dK[2],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[3] \text {$\#$1}+1\& ,3\right ]}dK[3],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[4] \text {$\#$1}+1\& ,4\right ]}dK[4],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[5] \text {$\#$1}+1\& ,5\right ]}dK[5],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.307 (sec), leaf count = 87
\[\left [y \left (x \right ) = \frac {5 \RootOf \left (1+8 \textit {\_Z}^{5}+\left (-2 x +2 \textit {\_C1} \right ) \textit {\_Z}^{2}\right )^{3}+2 \textit {\_C1} -2 x}{2 \RootOf \left (1+8 \textit {\_Z}^{5}+\left (-2 x +2 \textit {\_C1} \right ) \textit {\_Z}^{2}\right ) \left (4 \RootOf \left (1+8 \textit {\_Z}^{5}+\left (-2 x +2 \textit {\_C1} \right ) \textit {\_Z}^{2}\right )^{3}+\textit {\_C1} -x \right )}\right ]\] Mathematica raw input
DSolve[1 - y[x]*y'[x] + 3*y'[x]^5 == 0,y[x],x]
Mathematica raw output
{Solve[x + C[1] == Inactive[Integrate][Root[1 - K[1]*#1 + 3*#1^5 & , 1]^(-1), {K[1], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[1 - K[2]*#1 + 3*#1^5 & , 2]^(-1), {K[2], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[1 - K[3]*#1 + 3*#1^5 & , 3]^(-1), {K[3], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[1 - K[4]*#1 + 3*#1^5 & , 4]^(-1), {K[4], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[1 - K[5]*#1 + 3*#1^5 & , 5]^(-1), {K[5], 1, y[x]}], y[x]]}
Maple raw input
dsolve(3*diff(y(x),x)^5-y(x)*diff(y(x),x)+1 = 0, y(x))
Maple raw output
[y(x) = 1/2*(5*RootOf(1+8*_Z^5+(-2*x+2*_C1)*_Z^2)^3+2*_C1-2*x)/RootOf(1+8*_Z^5+(-2*x+2*_C1)*_Z^2)/(4*RootOf(1+8*_Z^5+(-2*x+2*_C1)*_Z^2)^3+_C1-x)]