ODE
\[ a y'(x)^3+b y'(x)^2+y'(x)^5=c y(x) \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 0.238324 (sec), leaf count = 216
\[\left \{\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [\text {$\#$1}^5+a \text {$\#$1}^3+b \text {$\#$1}^2-c K[1]\& ,1\right ]}dK[1],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [\text {$\#$1}^5+a \text {$\#$1}^3+b \text {$\#$1}^2-c K[2]\& ,2\right ]}dK[2],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [\text {$\#$1}^5+a \text {$\#$1}^3+b \text {$\#$1}^2-c K[3]\& ,3\right ]}dK[3],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [\text {$\#$1}^5+a \text {$\#$1}^3+b \text {$\#$1}^2-c K[4]\& ,4\right ]}dK[4],y(x)\right ],\text {Solve}\left [x+c_1=\int _1^{y(x)}\frac {1}{\text {Root}\left [\text {$\#$1}^5+a \text {$\#$1}^3+b \text {$\#$1}^2-c K[5]\& ,5\right ]}dK[5],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.143 (sec), leaf count = 40
\[\left [y \left (x \right ) = 0, x -\left (\int _{}^{y \left (x \right )}\frac {1}{\RootOf \left (\textit {\_Z}^{5}+a \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{2}-\textit {\_a} c \right )}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[b*y'[x]^2 + a*y'[x]^3 + y'[x]^5 == c*y[x],y[x],x]
Mathematica raw output
{Solve[x + C[1] == Inactive[Integrate][Root[-(c*K[1]) + b*#1^2 + a*#1^3 + #1^5 & , 1]^(-1), {K[1], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[-(c*K[2]) + b*#1^2 + a*#1^3 + #1^5 & , 2]^(-1), {K[2], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[-(c*K[3]) + b*#1^2 + a*#1^3 + #1^5 & , 3]^(-1), {K[3], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[-(c*K[4]) + b*#1^2 + a*#1^3 + #1^5 & , 4]^(-1), {K[4], 1, y[x]}], y[x]], Solve[x + C[1] == Inactive[Integrate][Root[-(c*K[5]) + b*#1^2 + a*#1^3 + #1^5 & , 5]^(-1), {K[5], 1, y[x]}], y[x]]}
Maple raw input
dsolve(diff(y(x),x)^5+a*diff(y(x),x)^3+b*diff(y(x),x)^2 = c*y(x), y(x))
Maple raw output
[y(x) = 0, x-Intat(1/RootOf(_Z^5+_Z^3*a+_Z^2*b-_a*c),_a = y(x))-_C1 = 0]