Internal problem ID [3154]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 408.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime } \left (4 x^{3}+\mathit {a1} x +\mathit {a0} \right )^{\frac {2}{3}}+\left (\mathit {a0} +\mathit {a1} y+4 y^{3}\right )^{\frac {2}{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 36
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), singsol=all)
\[ \int \frac {1}{\left (4 x^{3}+\mathit {a1} x +\mathit {a0} \right )^{\frac {2}{3}}}d x +\int _{}^{y \relax (x )}\frac {1}{\left (4 \textit {\_a}^{3}+\textit {\_a} \mathit {a1} +\mathit {a0} \right )^{\frac {2}{3}}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.502 (sec). Leaf size: 558
DSolve[y'[x](a0+a1 x+4 x^3)^(2/3)+(a0+a1 y[x]+4 y[x]^3)^(2/3)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\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}}=-\frac {3 \left (x-\text {Root}\left [4 \text {$\#$1}^3+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]\right ) \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 ]}} \, _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}}+c_1,y(x)\right ] \]