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 = 0.502506 (sec), leaf count = 211
\[\left \{\text {Solve}\left [c_1+x=\int _1^{y(x)} \frac {1}{\text {Root}\left [-c K[1]+\text {$\#$1}^5+\text {$\#$1}^3 a+\text {$\#$1}^2 b\& ,1\right ]} \, dK[1],y(x)\right ],\text {Solve}\left [c_1+x=\int _1^{y(x)} \frac {1}{\text {Root}\left [-c K[2]+\text {$\#$1}^5+\text {$\#$1}^3 a+\text {$\#$1}^2 b\& ,2\right ]} \, dK[2],y(x)\right ],\text {Solve}\left [c_1+x=\int _1^{y(x)} \frac {1}{\text {Root}\left [-c K[3]+\text {$\#$1}^5+\text {$\#$1}^3 a+\text {$\#$1}^2 b\& ,3\right ]} \, dK[3],y(x)\right ],\text {Solve}\left [c_1+x=\int _1^{y(x)} \frac {1}{\text {Root}\left [-c K[4]+\text {$\#$1}^5+\text {$\#$1}^3 a+\text {$\#$1}^2 b\& ,4\right ]} \, dK[4],y(x)\right ],\text {Solve}\left [c_1+x=\int _1^{y(x)} \frac {1}{\text {Root}\left [-c K[5]+\text {$\#$1}^5+\text {$\#$1}^3 a+\text {$\#$1}^2 b\& ,5\right ]} \, dK[5],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.109 (sec), leaf count = 40
\[ \left \{ x-\int ^{y \left ( x \right ) }\! \left ( {\it RootOf} \left ( {{\it \_Z}}^{5}+a{{\it \_Z}}^{3}+b{{\it \_Z}}^{2}-{\it \_a}\,c \right ) \right ) ^{-1}{d{\it \_a}}-{\it \_C1}=0,y \left ( x \right ) =0 \right \} \] 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] == 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] == 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] == 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] == 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] == 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),'implicit')
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