29.14.27 problem 408
Internal
problem
ID
[5006]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
14
Problem
number
:
408
Date
solved
:
Monday, January 27, 2025 at 10:02:43 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime } \left (4 x^{3}+\operatorname {a1} x +\operatorname {a0} \right )^{{2}/{3}}+\left (\operatorname {a0} +\operatorname {a1} y+4 y^{3}\right )^{{2}/{3}}&=0 \end{align*}
✓ 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)
\[
\int \frac {1}{\left (4 x^{3}+\operatorname {a1} x +\operatorname {a0} \right )^{{2}/{3}}}d x +\int _{}^{y \left (x \right )}\frac {1}{\left (4 \textit {\_a}^{3}+\textit {\_a} \operatorname {a1} +\operatorname {a0} \right )^{{2}/{3}}}d \textit {\_a} +c_{1} = 0
\]
✓ Solution by Mathematica
Time used: 14.692 (sec). Leaf size: 558
DSolve[D[y[x],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 ]
\]