Internal problem ID [3662]
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]
\[ \boxed {y^{\prime } \left (4 x^{3}+\operatorname {a1} x +\operatorname {a0} \right )^{\frac {2}{3}}+\left (\operatorname {a0} +\operatorname {a1} y+4 y^{3}\right )^{\frac {2}{3}}=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ \int \frac {1}{\left (4 x^{3}+\operatorname {a1} x +\operatorname {a0} \right )^{\frac {2}{3}}}d x +\int _{}^{y \left (x \right )}\frac {1}{\left (4 \textit {\_a}^{3}+\textit {\_a} \operatorname {a1} +\operatorname {a0} \right )^{\frac {2}{3}}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.581 (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 ]}} \operatorname {Hypergeometric2F1}\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 ]}} \operatorname {Hypergeometric2F1}\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 ] \]