4.9.11 \(\left (\text {a0}+\text {a1} x+4 x^3\right )^{2/3} y'(x)+\left (\text {a0}+\text {a1} y(x)+4 y(x)^3\right )^{2/3}=0\)

ODE
\[ \left (\text {a0}+\text {a1} x+4 x^3\right )^{2/3} y'(x)+\left (\text {a0}+\text {a1} y(x)+4 y(x)^3\right )^{2/3}=0 \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.663591 (sec), leaf count = 558

\[\text {Solve}\left [\frac {3 \left (\frac {x-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}} \left (x-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (x-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right )}{\left (\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (x-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]\right )}\right )}{\left (\text {a0}+\text {a1} x+4 x^3\right )^{2/3}}+\frac {3 \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]\right ) \left (\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}\right )^{2/3} \sqrt [3]{\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]\right )}{\left (\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]\right )}\right )}{\left (\text {a0}+\text {a1} y(x)+4 y(x)^3\right )^{2/3}}=c_1,y(x)\right ]\]

Maple
cpu = 0.105 (sec), leaf count = 36

\[ \left \{ \int \! \left (4\,{x}^{3}+{\it a1}\,x+{\it a0} \right ) ^{-{\frac {2}{3}}}\,{\rm d}x+\int ^{y \relax (x ) }\! \left (4\,{{\it \_a}}^{3}+{\it \_a}\,{\it a1}+{\it a0} \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(a0 + a1*y[x] + 4*y[x]^3)^(2/3) + (a0 + a1*x + 4*x^3)^(2/3)*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[(3*Hypergeometric2F1[1/3, 2/3, 4/3, ((x - Root[a0 + a1*#1 + 4*#1^3 & , 1])
*(-Root[a0 + a1*#1 + 4*#1^3 & , 2] + Root[a0 + a1*#1 + 4*#1^3 & , 3]))/((Root[a0
 + a1*#1 + 4*#1^3 & , 1] - Root[a0 + a1*#1 + 4*#1^3 & , 2])*(x - Root[a0 + a1*#1
 + 4*#1^3 & , 3]))]*(x - Root[a0 + a1*#1 + 4*#1^3 & , 1])*((x - Root[a0 + a1*#1 
+ 4*#1^3 & , 2])/(Root[a0 + a1*#1 + 4*#1^3 & , 1] - Root[a0 + a1*#1 + 4*#1^3 & ,
 2]))^(2/3)*((x - Root[a0 + a1*#1 + 4*#1^3 & , 3])/(Root[a0 + a1*#1 + 4*#1^3 & ,
 1] - Root[a0 + a1*#1 + 4*#1^3 & , 3]))^(1/3))/(a0 + a1*x + 4*x^3)^(2/3) + (3*Hy
pergeometric2F1[1/3, 2/3, 4/3, ((-Root[a0 + a1*#1 + 4*#1^3 & , 2] + Root[a0 + a1
*#1 + 4*#1^3 & , 3])*(-Root[a0 + a1*#1 + 4*#1^3 & , 1] + y[x]))/((Root[a0 + a1*#
1 + 4*#1^3 & , 1] - Root[a0 + a1*#1 + 4*#1^3 & , 2])*(-Root[a0 + a1*#1 + 4*#1^3 
& , 3] + y[x]))]*(-Root[a0 + a1*#1 + 4*#1^3 & , 1] + y[x])*((-Root[a0 + a1*#1 + 
4*#1^3 & , 2] + y[x])/(Root[a0 + a1*#1 + 4*#1^3 & , 1] - Root[a0 + a1*#1 + 4*#1^
3 & , 2]))^(2/3)*((-Root[a0 + a1*#1 + 4*#1^3 & , 3] + y[x])/(Root[a0 + a1*#1 + 4
*#1^3 & , 1] - Root[a0 + a1*#1 + 4*#1^3 & , 3]))^(1/3))/(a0 + a1*y[x] + 4*y[x]^3
)^(2/3) == C[1], y[x]]

Maple raw input

dsolve(diff(y(x),x)*(4*x^3+a1*x+a0)^(2/3)+(a0+a1*y(x)+4*y(x)^3)^(2/3) = 0, y(x),'implicit')

Maple raw output

Int(1/(4*x^3+a1*x+a0)^(2/3),x)+Intat(1/(4*_a^3+_a*a1+a0)^(2/3),_a = y(x))+_C1 = 
0